Prime numbers

1
Prime Numbers
SOLO HERMELIN
Updated: 28.10.12
: 12.09.13
: 05.03.15
http://www.solohermelin.com

2
SOLO
Table of Content
Primes
Euclid, Euclidean Division
Introduction
Prime Numbers
Euclid's Lemma
Fundamental Theorem of Arithmetic
Prime Numbers Formulas
Euler Zeta Function and the Prime History
Prime Number Distribution
Prime Number Theorem (PNT)
History of the Asymptotic Law of Distribution of Prime Numbers
The Chebychef Contribution
The Chebyschev Functions (1851)
The Chebyschev’s First Estimate
The Chebyschev’s Second Estimate
Riemann's Zeta Function (1859)
Riemann Zeta Function Zeros
Riemann's Zeta Function Properties
Von Mangoldt Psi Formula
Riemann's Zeta Function Relations
Abel’s Method of Partial Summation
( ) [ ]( ) 1
1 1 1
>
−
−=
−
− ∫
∞
+
σς xd
x
xx
s
s
s
s s
Möbius Function

3
SOLO
Table of Content (Continue – 1)
Primes
The Riemann Prime Number Formula
Hadamard Proof of the Prime Number Theorem (1896)
Newman’s Proof of the Prime Number Theorem (1980)
References
End of Presentation

4
SOLO
Table of Content (continue – 2)
Primes
Appendices
Definitions
Mellin Transform
Proof of Riemann's Zeta Function Relations
( ) ( ) ( ) 1Re
10
1
>=∞<
−
=Γ ∫
∞=
=
−
zxfordt
e
t
zz
t
t
t
z
ς
( )
( ) ( )
( )
∫
+∞=
−∞=
−
−
−
Γ
=
0
0
1
1sin2
1
i
i
z
d
e
i
zz
z
λ
λ
λ
λ
λ
π
ς
( ) ( ) ( )∫
+∞=
−∞=
−
−





=
−
−
0
0
1
1
2
sin22
1
i
i
z
z
z
z
id
e
λ
λ
λ
ς
π
πλ
λ
( ) ( ) ( )
∫
+∞=
−∞=
−
−
−−Γ
−=
0
0
1
12
1
i
i
z
d
ei
z
z
λ
λ
λ
λ
λ
π
ς
( ) ( ) ( ) ( ) ( )z
z
zzz
z
−





=Γ 1
2
sin22sin2 ς
π
πςπ
( ) ( ) ( ) ( )zzzz zz
−−Γ= −
112/sin2 1
ςππς
( ) ( )
( )
( )
( )[ ] ( )
( )
    
z
z
z
z
zzzz
−
−−−
−−Γ=Γ
1
2/12/
12/12/
ηη
ςπςπ
Bernoulli Numbers
Zeta-Function Values and the Bernoulli Numbers
Zeros of Zeta-Function: ζ (z) = 0
( ) ( ) ( ) 1,1ln
1
2ln
1
2ln
2
1
1
1
2
→−+
−
++
−
= ∑
∞
=
xasxn
nx
x
n
n
Oς

5
SOLO
Primes
Appendices
( ) 1
1
1
1
1
1
>+=





−== ∏∑
−
∞
=
σσς tis
pn
s
primep
s
n
s
Zeta Function ζ (s) and its Derivative ζ‘ (s)
( )
( )
( ) { } { } 1
1
1
>=+==− ∫
∞
−−
σσψ
ς
ς
tizduuuz
z
z
zd
d
z
ReRe
( ) ( )
( )∫
∞+
∞−
−
=
ic
ic
z
zd
z
x
z
z
i
x
ς
ς
π
ψ
'
2
1
( ) ( )
( ) 1,
1
ln
2
>+=
−
= ∫
∞
σσ
π
ς tisxd
xx
x
ss s
( ) ∑≤
=≤=
primep
xp
xprimesofnumberx 1:π
Hadamard Product of ζ (s)
Perron’s Formula
Auxiliary Tauberian Theorem
Infinite Series
Series of Functions
Absolute Convergence of Series of Functions
Uniformly Convergence of Sequences and Series

6
SOLO
Primes
Appendices
Infinite Products
The Mittag-Leffler and Weierstrass Theorems
The Weierstrass Factorization Theorem
The Hadamard Factorization Theorem
Mittag-Leffler’s Expansion Theorem
Generalization of Mittag-Leffler’s Expansion Theorem
Expansion of an Integral Function as an Infinite Product
The Hadamard Factorization Theorem
Hadamard Infinite Product Expansion of Zeta Function
Integration
Prime Number Applications

7
SOLO
Introduction
Primes
The start point of this presentation was the book of Marcus de Sautoy , “The
Music of the Primes”, 2003, Harper Collins Publisher, which I read during a
recreation trip to Crete. The subject was new for me, so to study this topic I
turned to the Internet, where I found many related articles. I spend a lot of time
trying to partially cover the subject, and this Presentation is the result.
It contains no original contributions, but clarifications, in my opinion, of some
of the topics.
In order to obtain a coherent presentation and complete some of the
proofs more work needs to be done
Return to TOC

8
SOLO Primes
Euclid
Euclid ( Eukleidēs), 300 BC, also known as Euclid of Alexandria,
was a Greek mathematician, often referred to as the "Father of
Geometry". He was active in Alexandria during the reign of Ptoleme I
(323–283 BC). His Elements is one of the most influential works in the
history of mathematics, serving as the main textbook for teaching
mathematics (especially geometry) from the time of its publication
until the late 19th or early 20th century.
In the Elements, Euclid
deduced the principles of what is now called Euclidean geometry from
a small set of axioms. Euclid also wrote works on perspective, conic
sections, spherical geometry, number theory and rigor.
Euclid" is the anglicized version of the Greek name Ε κλείδης,ὐ
meaning "Good Glory".
Euclid of Alexandria
Born: about 325 BC
Died: about 265 BC
in Alexandria, Egypt

9
SOLO Primes
Euclidean Division
In mathematics, and more particularly in arithmetic, the
Euclidean division is the usual process of division of integers
producing a quotient and a remainder. It can be specified precisely
by a theorem stating that these exist uniquely with given
properties.
Given two integers a and b, with b ≠ 0, there exist unique
integers q and r such that a = bq + r and 0 ≤ r < |b|, where |b|
denotes the absolute value of b
Statement of the Theorem
Proof
1. Existence
Statue of Euclid in the
Oxford University Museum
of Natural History
Consider first the case b < 0. Setting b' = −b and q' = −q, the equation a = bq + r may be rewritten a = b'q' + r
and the inequality 0 < r < |b| may be rewritten 0 < r < |b' |. This reduces the existence for the case b < 0 to that of
the case b > 0.
Similarly, if a < 0 and b > 0, setting a' = −a, q' = −q − 1 and r' = b − r, the equation a = bq + r may be rewritten
a' = bq' + r' and the inequality 0 < r < b may be rewritten 0 < r' < b. Thus the proof of the existence is reduced to
the case a ≥ 0 and b > 0 and we consider only this case in the remainder of the proof.
Let q1 and r1, both nonnegative, such that a = bq1 + r1, for example q1 = 0 and r1 = a. If r1 < b, we are done.
Otherwise q2 = q1 + 1 and r2 = r1 − b satisfy a = bq2 + r2 and 0 < r2 < r1. Repeating this process one gets eventually
q = qk and r = rk such that a = bq + r and 0 < r < b.
This proves the existence and also gives an algorithm to compute the quotient and the remainder. However this
algorithm needs q steps and is thus not efficient.

10
SOLO Primes
Euclidean Division
In mathematics, and more particularly in arithmetic, the
Euclidean division is the usual process of division of integers
producing a quotient and a remainder. It can be specified precisely
by a theorem stating that these exist uniquely with given
properties.
Given two integers a and b, with b ≠ 0, there exist unique
integers q and r such that a = bq + r and 0 ≤ r < |b|, where |b|
denotes the absolute value of b
Statement of the Theorem
Proof (continue)
2. Uniqueness
Statue of Euclid in the
Oxford University Museum
of Natural History
Suppose there exists q, q' , r, r' with 0 ≤ r, r' < |b| such that a = bq + r and a = bq' + r' .
Adding the two inequalities 0 ≤ r < |b| and −|b| < −r' ≤ 0 yelds −|b| < r − r' < |b|, that is |
r − r' | < |b|.
Subtracting the two equations yields: b(q' − q) = (r − r' ). Thus |b| divides |r − r' |. If |
r − r' | ≠ 0 this implies |b| < |r − r' |, contradicting previous inequality. Thus, r = r' and
b(q' − q) = 0. As b ≠ 0, this implies q = q' , proving uniqueness.
Return to TOC

11
SOLO Primes
Prime Numbers
Prime Number Definition:
A positive integer number p is prime if for all positive integers 1≤ a ≤p, we have for all the
Euclidean Divisions
p = a q + r
the reminder r = 0 only for (q=p, a=1) or (q=1, a=p).
A Prime Number is divisible only by 1 or by itself.
Proposition 20, Book IX of the Euclide’s Elements: “There are Infinitely many Primes”
Euclid's proof
Consider any finite set S of primes. The key idea is to consider the product of all
these numbers plus one:
∏∈
+=
Sp
pN 1
Like any other natural number, N is divisible by at least one prime number (it is
possible that N itself is prime).
None of the primes by which N is divisible can be members of the finite set S of
primes with which we started, because dividing N by any of these leaves a remainder
of 1. Therefore the primes by which N is divisible are additional primes beyond the
ones we started with. Thus any finite set of primes can be extended to a larger finite
set of primes.

12
SOLO Primes
Prime Numbers
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67
71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163
167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269
271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383
389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499
503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619
631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751
757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881
883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997
Here is a list of all the prime numbers up to 1,000:

13
SOLO Primes
Euclid's Lemma
In number theory, Euclid's lemma (also called Euclid's first theorem) is a lemma that
captures one of the fundamental properties of prime numbers. It states that if a prime
divides the product of two numbers, it must divide at least one of the factors. For example
since 133 × 143 = 19019 is divisible by 19, one or both of 133 or 143 must be as well. In
fact, 19 × 7 = 133. It is used in the proof of the fundamental theorem of arithmetic.
Let p be a prime number, and assume p divides the product of two integers a and b.
Then p divides a or p divides b (or perhaps both).
Divisibility Definition:
Assume a ≠ 0 and let b be any integer. If there is an integer q such
that b = a.
q, a is said to divide b; a is a divisor of b and b is a multiple
of a. Notation of a divide b is a|b.
The lemma first appears as proposition 30 in Book VII of Euclid's Elements. It is
included in practically every book that covers elementary number theory
Proof:
( ) ( ) ( ) 211221212211
222
111
rrrmrmpmmprpmrpmba
prrpmb
prrpma
⋅+++⋅=+⋅+=⋅
<+=
<+=
Using Euclidean Division Theorem
Since p|a.
b we must have r1
.
r2=0 meaning r1=0, or r2=0, or r1=0 and r2=0.
Return to TOC

14
SOLO Primes
In number theory, the fundamental theorem of arithmetic (also called the unique factorization
theorem or the unique-prime-factorization theorem) states (existence) that every integer greater
than 1 is either prime itself or is the product of prime numbers, and (uniqueness) that, although
the order of the primes in the second case is arbitrary, the primes themselves are not.
Book VII, propositions 30 and 32 of Euclid's Elements is essentially the statement and proof of the
fundamental theorem. Article 16 of Gauss' Disquisitiones Arithmeticae is an early modern
statement and proof employing modular arithmetic.
Canonical representation of a positive integer
Every positive integer n > 1 can be represented in exactly one way as a product of
prime powers:
∏=
==
k
i
ik
ik
ppppn
1
21
21 αααα

Proof of Fundamental Theorem of Arithmetic
Existence
By inspection, each of the small natural numbers 1, 2, 3, 4, ... is the product of primes. This is the
basis for a proof by induction. Assume it is true for all numbers less than n. If n is prime, there is
nothing more to prove. Otherwise, there are integers a and b, where n = ab and 1 < a ≤ b < n. By
the induction hypothesis, a = p1p2...pn and b = q1q2...qm are products of primes. But then n = ab =
p1p2...pnq1q2...qm is the product of primes

15
SOLO Primes
Canonical representation of a positive integer
Every positive integer n > 1 can be represented in exactly one way as a product of
prime powers:
∏=
==
k
i
ik
ik
ppppn
1
21
21 αααα

Proof of Fundamental Theorem of Arithmetic (continue)
Uniqueness
Assume that s > 1 is the product of prime numbers in two different ways:
nm qqqppps  2121 ==
We must show m = n and that the qj are a rearrangement of the pi.
By Euclid's lemma p1 must divide one of the qj; relabeling the qj if necessary, say that p1 divides
q1. But q1 is prime, so its only divisors are itself and 1. Therefore, p1 = q1, so that
nm qqpp
p
s
 22
1
==
This can be done for all m of the pi, showing that m ≤ n. If there were any qj left over we would
have
which is impossible, since the product of numbers greater than 1 cannot equal 1.
Therefore m = n and every qj is a pi.
nm
m
qq
ppp
s


1
21
1 +==
q.e.d.
Return to TOC

16
SOLO Primes
Sieve of Eratosthenes
Eratosthenes of Cyrene
( c. 276 BC – c. 195/194 BC)
The sieve of Eratosthenes (Greek: κόσκινον ρατοσθένους),Ἐ
one of a number of prime number sieves, is a simple, ancient
algorithm for finding all prime numbers up to any given limit.
It does so by iteratively marking as composite (i.e., not prime)
the multiples of each prime, starting with the multiples of 2
The multiples of a given prime are generated as a
sequence of numbers starting from that prime, with
constant difference between them that is equal to that
prime.[1]
This is the sieve's key distinction from using
trial division to sequentially test each candidate number
for divisibility by each prime.[2]
The sieve of Eratosthenes is one of the most efficient
ways to find all of the smaller primes. It is named after
Eratosthenes of Cyrene, a Greek mathematician;
although none of his works has survived, the sieve was
described and attributed to Eratosthenes in the
Introduction to Arithmetic by Nicomachus.
Sieve of Eratosthenes: algorithm steps for
primes below 121 (including optimization
of starting from prime's square
http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes
Return to TOC

17
SOLO Primes
Marin Mersenne,
Marin Mersennus or
le Père Mersenne
(1588 –1648)
Mersenne Prime
In mathematics, a Mersenne number, named after Marin Mersenne
(a French monk who began the study of these numbers in the early
17th century), is a positive integer that is one less than a power of
two:
12 −= p
pM
Named after Marin Mersenne
Publication year 1636[1]
Author of publication Regius, H.
Number of known terms 47
Conjectured number of
terms
Infinite
Subsequence of Mersenne numbers
First terms 3, 7, 31, 127
Largest known term 243112609
− 1
OEIS index A000668
As of October 2009[ref]
, 47 Mersenne primes are known. The
largest known prime number (243,112,609
– 1) is a Mersenne prime.[3]
Since 1997, all newly-found Mersenne primes have been
discovered by the "Great Internet Mersenne Prime Search"
(GIMPS), a distributed computing project on the Internet.
A basic theorem about Mersenne numbers states that in order for
Mp to be a Mersenne prime, the exponent p itself must be a prime
number. This rules out primality for numbers such as
M4 = 24
− 1 = 15: since the exponent 4 = 2×2 is composite, the
theorem predicts that 15 is also composite; indeed, 15 = 3×5
While it is true that only Mersenne numbers Mp, where
p = 2, 3, 5, … could be prime - and it was believed by early
mathematicians that all such numbers were prime[2]
- Mp is very
rarely prime even for a prime exponent p. The smallest
counterexample is the Mersenne number
89x2320471211
11 ==−=M

18
SOLO Primes
Goldbach’s Conjecture
Christian Goldbach
(1690 –1764)
Goldbach's conjecture is one of the oldest and best-known unsolved
problems in number theory and in all of mathematics. It states:
Every even integer greater than 2 can be expressed as the sum
of two primes.
The conjecture has been shown to be correct[2]
up through 4 × 1018
and is
generally assumed to be true, but no mathematical proof exists despite
considerable effort
History:
On 7 June 1742, the German mathematician Christian Goldbach
(originally of Brandenburg-Prussia) wrote a letter to Leonhard
Euler (letter XLIII)[4]
in which he proposed the following
conjecture:
Every integer which can be written as the sum of two
primes, can also be written as the sum of as many primes as
one wishes, until all terms are units
He then proposed a second conjecture in the margin of his letter
Every integer greater than 2 can be written as the sum of three primes
The two conjectures are now known to be equivalent, but this did not seem to be an issue at the
time
Return to TOC

SOLO Primes
Euler Zeta Function and the Prime History
++++ 232
4
1
3
1
2
1
1
In 1650 Mengoli asked if a solution exists for
P. Mengoli
1626 - 1686
The problem was tackled by Wallis, Leibniz, Bernoulli family, without success.
The solution was given by the young Euler in 1735. The problem was named “Basel
Problem” for Basel the town of Bernoulli and Euler.
Euler started from Taylor series expansion of the sine function
+−+−=
!7!5!3
sin
753
xxx
xx
Dividing by x, he obtained
+−+−=
!7!5!3
1
sin 642
xxx
x
x
The roots of the left side are x =±π, ±2π, ±3π,…. However sinx/x is not a
polynomial, but Euler assumed (and check it by numerical computation)
that it can be factorized using its roots as
 ⋅





−⋅





−⋅





−=





+⋅





−⋅





+⋅





−= 2
2
2
2
2
2
9
1
4
11
2
1
2
111
sin
πππππππ
xxxxxxx
x
x
Leonhard Euler
(1707 – 1783)

SOLO Primes
+−+−=
!7!5!3
1
sin 642
xxx
x
x ⋅





−⋅





−⋅





−= 2
2
2
2
2
2
9
1
4
11
sin
πππ
xxx
x
x
Leonhard Euler
(1707 – 1783)If we formally multiply out this product and collect all the x2
terms, we
see that the x2
coefficient of sin(x)/x is
∑
∞
=
−=





+++−
1
22222
11
9
1
4
11
n nππππ

But from the original infinite series expansion of sin(x)/x, the coefficient of x2
is
−1/(3!) = −1/6. These two coefficients must be equal; thus,
∑
∞
=
−=−
1
22
11
6
1
n nπ 6
1 2
1
2
π
=∑
∞
=n n
Euler extend this to a general function, Euler Zeta Function
( )  ,4,3,2
4
1
3
1
2
1
1: =++++= nn nnn
ς
The sum diverges for n ≤ 1 and
converges for n > 1.
Euler computed the sum for n up to n = 26. Some of the values are given here
( ) ( ) ( ) ( ) ,
9450
8,
945
6,
90
4,
6
2
8642
π
ς
π
ς
π
ς
π
ς ====
Euler checked the sum
for a finite number of
terms.
Euler Zeta Function and the Prime History (continue – 1)

SOLO Primes
Euler Product Formula for the Zeta Function
Leonhard Euler proved the Euler product formula for the Riemann
zeta function in his thesis Variae observationes circa series infinitas
(Various Observations about Infinite Series), published by St
Petersburg Academy in 1737
∏∑ −
∞
= −
=
primep
x
n
x
pn 1
11
1
where the left hand side equals the Euler Zeta Function
Euler Proof of the Product Formula
( ) ++++= xxxxx
s
8
1
6
1
4
1
2
1
2
1
ς
( ) +++++++=





− xxxxxxx
x
13
1
11
1
9
1
7
1
5
1
3
1
1
2
1
1 ς
( ) ++++++=





− xxxxxxxx
x
33
1
27
1
21
1
15
1
9
1
3
1
2
1
1
3
1
ς
( ) ++++++=





−





− xxxxxxx
x
17
1
13
1
11
1
7
1
5
1
1
2
1
1
3
1
1 ς
all elements having a factor of 3 or 2 (or both) are removed
( ) +++++== ∑
∞
=
xxxx
n
x
n
x
5
1
4
1
3
1
2
1
1
1
1
ς converges for integer x > 1
all elements having a
factor of 2 are
removed
Leonhard Euler
(1707 – 1`783)
EulerZeta Function and the Prime History (continue – 2)

SOLO Primes
Leonhard Euler
(1707 – 1`783)
Euler Product Formula for the Zeta Function
( ) ∏∑ −
∞
= −
==
primep
x
n
x
pn
x
1
11
1
ς
Euler Proof of the Product Formula (continue)
( ) ++++++=





−





− xxxxxxx
x
17
1
13
1
11
1
7
1
5
1
1
2
1
1
3
1
1 ς
Repeating infinitely, all the non-prime elements are removed, and we get:
( ) 1
2
1
1
3
1
1
5
1
1
7
1
1
11
1
1
13
1
1
17
1
1 =





−





−





−





−





−





−





− xxxxxxxx
ς
Dividing both sides by everything but the ζ(s) we obtain
( )






−





−





−





−





−





−
=
xxxxxx
x
13
1
1
11
1
1
7
1
1
5
1
1
3
1
1
2
1
1
1
ς
Therefore
( ) ∏∑ −
∞
= −
==
primep
x
n
x
pn
x
1
11
1
ς

SOLO Primes
Leonhard Euler
(1707 – 1`783)
Euler Product Formula for the Riemann Zeta Function
( ) ∏∑ −
∞
= −
==
primep
s
n
s
pn
s
1
11
1
ς
Another Proof:
According to Fundamental Theorem of Arithmetic: Every
positive integer n > 1 can be represented by exactly one way as
a product of prime powers
integer,21
21 −−= iik primeppppn k
α
ααα

( ) ( )∑∑
∞
=
−−−
∞
=
==
1
21
1
21
1
n
s
k
n
s
k
ppp
n
s
ααα
ς 
( ) ( ) ∏∏ ∑∑∑ −
∞
=
−
∞
=
−−−
∞
= −
====
primep
s
primep k
sk
n
s
k
n
s
p
pppp
n
s k
1
11
11
21
1
21 ααα
ς 
Since in the sum n covers all the integers, for each prime there are the
powers of al integers k ϵ [1,∞)

24
SOLO Primes
The Euler zeta function, ζ(s), is a function is the sum of the infinite series
( ) ∑
∞
=
=
1
1
n
x
n
xς
Let compute






=
≠
+−=
∞
∞+−
∞
−
∫
1,ln
1,
1
1
1
1
1
px
p
s
x
dxx
s
s
According to Maclaurin – Euler Integral Convergence Test for Infinite Series
the integral and therefore the series are divergent for p ≤ 1, convergent for p > 1.
Leonhard Euler
(1707 – 1`783)
Euler Zeta Function and the Prime History (continue – 5)
Euler Zeta Function for x > 1
( )
( )
( )
( )
( )
( ) 0823.1
90
1
2
1
14
202.1
1
2
1
13
645.1
6
1
2
1
12
612.22/3
1
2
1
11
2
1
0
4
44
33
2
22
≈=++++=
≈++++=
≈=++++=
≈
∞=++++=
−=
π
ς
ς
π
ς
ς
ς
ς




n
n
n
n

SOLO Primes
Euler Product Formula
( ) ∏∑ −
∞
= −
==
primep
s
n
s
pn
s
1
11
1
ς
Another Proof of the Product Formula
Start with the following geometric series expansion
 ++++++=
− − skssss
ppppp
1111
1
1
1
32
When , we have |p−s
| < 1 and this series converges absolutely
Hence we may take a finite number of factors, multiply them together, and
rearrange terms. Taking all the primes p up to some prime number limit q, we
have
( ) ∑∏
∞
+=≤
−
<
−
−
1
1
1
1
qsqp
s
np
s σ
ς
where σ is the real part of s. By the fundamental theorem of arithmetic, the partial
product when expanded out gives a sum consisting of those terms n−s
where n is a
product of primes less than or equal to q. The inequality results from the fact that
therefore only integers larger than q can fail to appear in this expanded out partial
product. Since the difference between the partial product and ζ(s) goes to zero when
σ > 1, we have convergence in this region.
Leonhard Euler
(1707 – 1`783)
Return to TOC

26
In number theory, the Prime Number Theorem (PNT) describes the asymptotic
distribution of the prime numbers. The prime number theorem gives a general
description of how the primes are distributed amongst the positive integers.
Prime Number Distribution
SOLO Primes
Since a general formula for the Prime determination couldn’t be found, the
attention was driven to the following question:
How to find a function that defines the number of primes less or equal to a given
number x? This function was named π (x)
( ) ∑≤
=≤=
primep
xp
The first question that was unsuccessful tackled was:
Given a integer number N, how to find the Prime Number P, less then N, and as
closed as possible to N.
Return to TOC

27
In number theory, the Prime Number Theorem (PNT) describes the asymptotic
distribution of the prime numbers. The prime number theorem gives a general
description of how the primes are distributed amongst the positive integers.
Let π(x) be the prime-counting function that gives the number of primes less than
or equal to x, for any real number x. For example, π(10) = 4 because there are four
prime numbers (2, 3, 5 and 7) less than or equal to 10. The Prime Number Theorem
then states that the limit of the quotient of the two functions π(x) and x / ln(x) as x
approaches infinity is 1, which is expressed by the formula
( )
( )
1
ln/
lim =
∞→ xx
x
x
π
π(x)
x / ln(x)
SOLO Primes
Return to TOC

28
SOLO Primes
Based on the tables by Anton Felkel and Jurij Vega, Adrien-Marie
Legendre conjectured in 1797 or 1798 that π(a) is approximated by the
function a/(A ln(a) + B), where A and B are unspecified constants. In
the second edition of his book on number theory (1808) he then made
a more precise conjecture, with A = 1 and B = −1.08366.
Adrien-Marie Legendre
)1752–1833(
Carl Friedrich Gauss considered the same question: "Ins Jahr 1792
oder 1793", according to his own recollection nearly sixty years later
in a letter to Encke (1849), he wrote in his logarithm table (he was
then 15 or 16) the short note "Primzahlen unter
But Gauss never published this conjecture.
( )
BaA
a
a
+
≈
ln
π
( )
a
a
a
ln
≈π

29
SOLO Primes
Later Gauss came up with a new approximating function, the
logarithmic integral Li (x)
( ) ∫=
x
u
du
xLi
2
ln
:
Calculating ( ) ( ) ( )
1000
1000−−
=∆
xx
x
ππ
Computing by hand, it seams that Δ(x) tends to zero ,but very slowly. To see how
slow computing the inverse of Δ(x) it was found that
( ) xx ln/1 ≈∆
Meaning that ( )
x
x
ln
1
≈∆
Define
Carl Friedrich Gauss
(1777 – 1855)
( )xLi
( )xπ
x
x
ln

30
x π(x( π(x( − x / ln x π(x( / (x / ln x( li(x( − π(x( x / π(x(
10 4 −0.3 0.921 2.2 2.500
102
25 3.3 1.151 5.1 4.000
103
168 23 1.161 10 5.952
104
1,229 143 1.132 17 8.137
105
9,592 906 1.104 38 10.425
106
78,498 6,116 1.084 130 12.740
107
664,579 44,158 1.071 339 15.047
108
5,761,455 332,774 1.061 754 17.357
109
50,847,534 2,592,592 1.054 1,701 19.667
1010
455,052,511 20,758,029 1.048 3,104 21.975
1011
4,118,054,813 169,923,159 1.043 11,588 24.283
1012
37,607,912,018 1,416,705,193 1.039 38,263 26.590
1013
346,065,536,839 11,992,858,452 1.034 108,971 28.896
1014
3,204,941,750,802 102,838,308,636 1.033 314,890 31.202
1015
29,844,570,422,669 891,604,962,452 1.031 1,052,619 33.507
1016
279,238,341,033,925 7,804,289,844,393 1.029 3,214,632 35.812
1017
2,623,557,157,654,233 68,883,734,693,281 1.027 7,956,589 38.116
1018
24,739,954,287,740,860 612,483,070,893,536 1.025 21,949,555 40.420
1019
234,057,667,276,344,607 5,481,624,169,369,960 1.024 99,877,775 42.725
1020
2,220,819,602,560,918,840 49,347,193,044,659,701 1.023 222,744,644 45.028
1021
21,127,269,486,018,731,928 446,579,871,578,168,707 1.022 597,394,254 47.332
1022
201,467,286,689,315,906,290 4,060,704,006,019,620,994 1.021 1,932,355,208 49.636
1023
1,925,320,391,606,803,968,923 37,083,513,766,578,631,309 1.020 7,250,186,216 51.939
SOLO Primes

31
SOLO Primes
Both Gauss's formulas imply the same conjectured asymptotic
equivalence of π(x) , x / lnx and Li (x) stated above, although it
turned out that Gauss's Li (x) approximation is considerably better
if one considers the differences instead of quotients. By using
L’Hopital theorem we can see that
( )
( )
( )
1
1ln
ln
lim
ln
1ln
ln
1
lim
ln
lim
ln/
lim
2
=
−
=
−
=






=
∞→∞→∞→∞→ x
x
x
x
x
x
x
xd
d
xLi
xd
d
xx
xLi
xxxx
Example:
( ) ( ) ( )[ ] ( )[ ] 6115ln/,128,78498,106
=−=−== nnnnnLinn πππ
(1777 – 1855)

32
SOLO Primes
Gauss's function compared to the true number of
primes
Gauss's guess was based on throwing a dice with one side marked "prime" and the
others all blank. The number of sides on the dice increases as we test larger numbers
and Gauss discovered that the logarithm function could tell him the number of sides
needed. For example, to test primes around 1,000 requires a six-sided dice. To make his
guess at the number of primes, Gauss assumed that a six-sided dice would land exactly
one in six times on the prime side. But of course it is very unlikely that a dice thrown
6,000 times will land exactly 1,000 times on the prime side. A fair dice is allowed to
over- or under-estimate this score. But was there any way to understand how to get from
Gauss's theoretical guess to the way the prime number dice had really landed? Aged 33,
Riemann, now working in Göttingen, discovered that music could explain how to
change Gauss's graph into the staircase graph that really counted the primes.
(1777 – 1855)
University of Göttingen

33
SOLO Primes
John Edensor Littlewood
1885 - 1977
( ) ( )( )
xx
xxLix
lnlnln
ln
2/1
−π
.10
3410
10
<x
.10
310
10
<x
Gauss asserted that π (x) < Li (x). Toward the end of his 1859 paper
Riemann makes the same assertion. Using computation this was
proved to be true for all x < 108
.
In 1914 Litlewood showed that π (x) – Li (x) changes sign infinitely
often. He showed that there is a constant K > 0 such that
is greater than K for arbitrarily large x and less than –K for arbitrarily large x.
Litlewood’s method helped Skewes, who in 1933, showed that there is at least one sign
change at x for some
Skewes proof required the Riemann Hypothesis. In 1955 he obtained a bound without
using the Riemann Hypothesis. This new bound was
Skewes large bound can be reduced substantially. In 1966 Sherman Leham showed
that between 1.53x101165
and 1.65x101165
there are more than 10500
successive integers x
for which π (x) > Li (x). Lehman work suggest there is no sign change before 1020
.
In 1987 Riele showed that between 6.62x10370
and 6.69x10370
there are more than 10180
successive integers for which π (x) > Li (x).

34
SOLO Primes
In 1837 Johann Peter Gustav Lejeune Dirichlet introduced
Dirichlet Series
Johann Peter Gustav Lejeune
Dirichlet
)1805–1859(
( ) ( )
∑
∞
=
=
1
:ˆ
n
s
n
nf
sf
is convergent for Re (s) > c if f (n) = O (n c-1
) as n → ∞.
Given the Perron’s Formula
Oskar Perron
( 1880 – 1975)
0
11
10
2
1
>



>
<
=∫
∞+
∞−
ε
π
ε
ε
xif
xif
ds
s
x
i
i
i
n
then
( ) ( ) ( ) ( )∑∑∫ ∑∫ ≤≤
∞
=
∞+
∞−
∞
=
∞+
∞−
=



>
<
⋅==
xnn
i
i n
s
s
i
i
s
nf
nxif
nxif
nf
s
ds
n
nf
x
is
ds
sfx
i 111 1
0
2
1ˆ
2
1
ε
ε
ε
ε
ππ
For f (n) = 1 we obtain the Zeta Function ( ) ∑
∞
=
=
1
1
:
n
s
n
sς
therefore
( ) ∑∫ ≤≤
∞+
∞−
=
xn
i
i
s
s
ds
sx
i 1
1
2
1
ε
ε
ς
π

SOLO Primes
In two papers from 1848 and 1850, the Russian mathematician
Pafnuty L'vovich Chebyshev attempted to prove the asymptotic
law of distribution of prime numbers. His work is notable for the
use of the zeta function ζ(s) (for real values of the argument
"s", as are works of Leonhard Euler, as early as 1737)
predating Riemann's celebrated memoir of 1859, and he
succeeded in proving a slightly weaker form of the asymptotic
law, namely, that if the limit of π(x)/(x/ln(x)) as x goes to infinity
exists at all, then it is necessarily equal to one.[2]
He was able to
prove unconditionally that this ratio is bounded above and below
by two explicitly given constants near to 1 for all x.[3]
Leonhard Euler
(1707 – 1`783)
Joseph Louis François
Bertrand
(1822 –1900)
Although Chebyshev's paper did not prove the Prime Number
Theorem, his estimates for π(x) were strong enough for him to prove
Bertrand's postulate that there exists a prime number between n and
2n for any integer n ≥ 2.
( ) 5/6,30/532log 12
30/15/13/12/1
1 ccc ==where , and N is sufficiently large.
( ) ( ) ( )
N
N
cN
N
N
c
lnln
11 επε +≤≤−
Pafnuty Lvovich
Chebyshev
) )1821–1894

SOLO Primes
Without doubt, the single most significant paper concerning
the distribution of prime numbers was Riemann's 1859
memoir On the Number of Primes Less Than a Given
Magnitude, the only paper he ever wrote on the subject.
Riemann introduced revolutionary ideas into the subject, the
chief of them being that the distribution of prime numbers is
intimately connected with the zeros of the analytically
extended Riemann zeta function of a complex variable. In
particular, it is in this paper of Riemann that the idea to apply
methods of complex analysis to the study of the real function
π(x) originates. Extending these deep ideas of Riemann, two
proofs of the asymptotic law of the distribution of prime
numbers were obtained independently by Jacques Hadamard
and Charles Jean de la Vallée-Poussin and appeared in the
same year (1896). Both proofs used methods from complex
analysis, establishing as a main step of the proof that the
Riemann zeta function ζ(s) is non-zero for all complex values
of the variable s that have the form s = 1 + i t with t > 0
Georg Friedrich Bernhard
Riemann
)1826–1866(
Jacques Salomon
Hadamard
(1865 –1963)
Charles-Jean Étienne Gustave Nicolas
de la Vallée Poussin
(1866 1962)

SOLO Primes
During the 20th century, the theorem of Hadamard and de la
Vallée-Poussin also became known as the Prime Number
Theorem. Several different proofs of it were found, including
the "elementary" proofs of Atle Selberg and Paul Erdős (1949).
While the original proofs of Hadamard and de la Vallée-
Poussin are long and elaborate, and later proofs have
introduced various simplifications through the use of
Tauberian theorems but remained difficult to digest, a
surprisingly short proof was discovered in 1980 by American
mathematician Donald J. Newman. Newman's proof is
arguably the simplest known proof of the theorem, although it
is non-elementary in the sense that it uses Cauchy's integral
theorem from complex analysis
Atle Selberg
(1917 –2007)
Paul Erdős
(1913 –1996)
Donald J. Newman
( 1930 –2007)
Return to TOC

38
SOLO Primes
The Chebychef Contribution
integeres,
1
21
21
−−== ∏=
ii
m
i
k
i
k
m
kk
kprimespppppn im

The starting point is that any positive number can be factored into a
unit product of powers of distinct primes
integeres,lnlnlnlnln
1
2211 −−=+++= ∑=
ii
m
i
iimm kprimesppkpkpkpkn 
The utility of this formula is enhanced by the use of von Mangold
symbol Λ (n)
( )


 >=
=Λ
otherwise
kandpprimesomeforpnifp
n
k
0
1integerln
Hans Carl Friederich
von Mangold
(1854 – 1925)
The symbol Σj|n will be used to denote a sum on j where j runs through all of the
positive divisors of the positive integer n. With this notation we have:
( ) ∑∑ =
=Λ=
m
i
ii
nj
pkjn
1|
lnln
To prove this note that from and the definition of Λ (j) the
only nonzero terms that can appear on the right side are ln p1,ln p2,…,ln pk.
Moreover p1 appears for j=p1, j=p1
2
,…,j=p1
k1
. Thus ln p1 appears exactly
k times. Similarly p appears exactly k times, etc
mk
m
kk
pppn 21
21=
Since we have products a most useful formula is obtained by using
natural logarithm
Pafnuty Lvovich
Chebyshev
) )1821–1894
Return to TOC

39
SOLO Primes
( )


 >=
=Λ
otherwise
n
k
0
1integerln Von Mangoldt Function
1895
Pafnuty Lvovich
Chebyshev
) )1821–1894
The Chebyschev Functions (1851)
( ) ∑≤
=
primep
xp
px ln:θ
Chebyschev Theta Function
( ) ( ) ∑∑
≤≤
=Λ=
primep
xpxn k
pnx ln:ψ
Chebyschev Psi Function
From the definition of Chebyschev Psi Function and
of Λ (j)
( ) ( )
( ) ( ) ( ) 

+++=
=+++=Λ= ∑∑∑∑∑
≤≤≤≤≤
3/12/1
lnlnlnln:
32
xxx
ppppnx
primep
xp
primep
xp
primep
xp
primep
xpxn k
θθθ
ψ

40
SOLO Primes
( ) 3ln2ln7ln5ln2ln3ln2lnln ++++++== ∑
≤
primep
xpk
pxψ
( )


 >=
=Λ
otherwise
n
k
0
1integerln
The Chebyschev Functions (continue - 1)
( ) 7ln5ln3ln2lnln:10
10
+++=== ∑≤
primep
p
pxθ
Example: x = 10
Prime Numbers p < x = 10 :
p: 2, 3, 5, 7
Prime Numbers p2
< x = 10 :
p2
: 22
=4, 32
=9
Prime Numbers p3
< x = 10 :
p3
: 23
=8,
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ,010,3ln9,2ln8,7ln7,06
,5ln5,2ln4,3ln3,2ln2,01
=Λ=Λ=Λ=Λ=Λ
=Λ=Λ=Λ=Λ=Λ
( ) ( ) 7ln5ln3ln22ln3:10
10
++⋅+⋅=Λ== ∑=≤xn
nxψ
1621028 43
=<=<= x




=→<<
2ln
10ln
32ln410ln2ln3
[ ] [ ] 10..integral: <−<= xxtsx
( ) ∑∑ ≤≤






==
primep
xp
primep
xp
p
p
x
px
k
ln
ln
ln
lnψ

41
SOLO Primes
( ) ( ) ( ) ( ) ( ) +++==Λ= ∑∑
≤≤
3/2/ln: xxxpnx
primep
xpxn k
θθθψ
( )


 >=
=Λ
otherwise
n
k
0
1895
( ) ∑≤
=
primep
xp
px ln:θ
Theorem ( ) ( ) ( )
x
x
x
x
xx
x
xxx
ψθπ
∞→∞→∞→
== limlim
ln/
lim
Proof:
( ) ∑≤
=≤=
primep
xp
( ) ( ) ( ) ( ) ( ) ∑∑ ≤
≤
≤
≤=+++=≤
primep
xp
xp
primep
xp
xpxxxxx
k
1lnln3/2/
lnln
θθθψθ
Define:
( ) ( ) ( ) 11
ln
ln/
::,: 321 >==== ∑≤
x
x
x
xx
x
L
x
x
L
x
x
L
primep
xp
πψθ
Therefore: 321 LLL ≤≤
One the other hand, if 0 < α <1, x > 1, then: x > α → ln x > ln α
( ) ( ) ( )[ ]
( )
( )[ ] xxxxxxxxppx
xx
xpxxpx
xp
xp
xpx
primep
xp
lnlnln1lnlnln:
10lnln
α
π
α
αα
παππααθ
αα
αα
α
α
α
−≥−=








=≥≥=
≤
<<
≤<≤<
≥
≥
≤<≤
∑∑∑∑
Return to Newman
Proof of PNT
Chebyshev didn’t prove that the limit is 1.

42
SOLO Primes
( ) ( ) ( ) ( ) ( ) +++==Λ= ∑∑
≤≤
3/2/ln: xxxpnx
primep
xpxn k
θθθψ
( ) ∑≤
=
primep
xp
px ln:θ
Theorem
( ) ( ) ( )
x
x
x
x
xx
x
xxx
ψθπ
∞→∞→∞→
== limlim
ln/
lim
Proof (continue):
( ) ∑≤
=≤=
primep
xp
Define:
( ) ( ) ( ) 11
ln
ln/
::,: 321 >==== ∑≤
x
x
x
xx
x
L
x
x
L
x
x
L
primep
xp
πψθ
321 LLL ≤≤
( ) ( )[ ] xxxx lnα
παθ −≥
Dividing the inequality by x > 1 we obtain:
( ) ( )




−
⋅
≥ −α
π
α
θ
1
lnln
x
x
x
xx
x
x
Keep α fixed and x → ∞ we obtain: 0
ln
lim
10
1
<<
−∞→
=
α
α
x
x
x
Hence:
( ) ( )
31 lim
ln
limlimlim L
x
xx
L
x
x
xxxx ∞→∞→∞→∞→
=
⋅
≥= α
π
α
θ
gives: ( ) ( ) ( )321 limlimlim LLL
xxx ∞→∞→∞→
== q.e.d.
Tacking α→1: 31 limlim LL
xx ∞→∞→
≥ together with 321 LLL ≤≤
Return to TOC
( ) ( )xx O=ψReturn to

43
SOLO Primes
( ) ( )xx O=θ
The Chebyschev’s First Estimate ( ) ∑≤
=
primep
xp
px ln:θ
Theorem
Proof: Start with the Binomial formula
( )

( ) ( )( )
( ) 121
1212222
112
integer
2
0
22
⋅−
++−
=





≥





=+= ∑= 

nn
nnnn
n
n
k
nn
k
nn
( ) ( )nn
pppp
npn
eeeep
n
n npnpnpnnpn θθ −
−∏
<<
=
∑∑
=
∑
==≥




 <<<<<<
∏ 2
lnlnlnln
2
222
2
Taking natural algorithm from both sides, we obtain ( ) ( )nnn θθ −≥ 22ln2
Definition of O:
We say that f (x) = O (g (x)) if exists a
constant k > 0 such that |f (x)| < k |g (x)|
( ) ( )( ) ∏∏∏∏ <<<<<<<<
==≥=++−=
nk
primep
npnnpnnkn
kbbydividednotispcpknnnna
1222
:&:12122 
c
b
a
pkcbka
npnnknkn
≥⋅=⋅≥= ∏∏∏ <<<<<< 212
:

44
SOLO Primes
( ) ( )xx O=θ
The Chebyschev’s First Estimate
( ) ∑≤
=
primep
xp
px ln:θ
Theorem
Proof (continue):
q.e.d.
Definition of O:
Let be r the minimal integer such that 2r
> x. Then
( ) ( ) ( ) xxx 2ln12/ +≤−θθ
( ) ( ) ( )
  

  






−





+





−−





+





−





+





−=





−= ++ rrrr
xxxxxxx
x
x
xx
22222222 1122
θθθθθθθθθθθ
Therefore ( ) ( )xx O=θ
( ) ( ) ( ) xx
xxx rr
j
j
r
j
jj
2ln12
2
1
1
2
1
1
2ln1
2
2ln1
22
1
0
1
0
1
+≤
−
−
+=+≤











−





= ∑∑
−
=
−
=
+
θθ
Taking natural algorithm from both sides, we obtain ( ) ( )nnn θθ −≥ 22ln2
Define [x] the biggest integer less than x; i.e. 0 < x – [x] < 1 Then
( ) ( ) ( ) ( ) ( ) x
x
x
xxx
x
x
xxx 2ln12ln
2
2ln
22
2
2
2
2
2/ +≤



+≤









−









+









−=









−=−
    
θθθθθθθθ
Return to TOC

45
SOLO Primes
( ) ( )xx O=ψ
( ) ∑≤
=
primep
xp
px ln:θ
Theorem
Proof:
Definition of O:
( ) ( ) ( ) ( ) ( ) +++==Λ= ∑∑
≤≤
3/2/ln: xxxpnx
primep
xpxn k
θθθψ
For 0 < δ < 1 and y = x1-δ
, we have
( ) ( )
( ) ( ) ( ) ( )
x
x
x
y
x
yyx
y
x
p
y
p
y
yy
primep
xpy
primep
xp
primep
xpy
primep
xpy
primep
yp
ln1
1
ln
1
ln
ln
ln
1
ln
ln
1
1&1
1 θ
δ
θ
ππ
θ
π
δ
−
+=+≤+=
=≤≤≤=
−
≤<
≤≤<≤<≤
∑
∑∑∑∑
Therefore
( ) ( ) ( ) ( ) ( )
x
x
x
x
x
x
x
x
xx
x
x
x
x
x ψ
δ
θ
δ
πψθ
δδ
⋅
−
+≤⋅
−
+≤≤≤
1
1ln
1
1ln
ln/
We also proved that ( ) ( ) ( )
xx
x
x
x
x
x
ln/
πψθ
≤≤

46
SOLO Primes
( ) ( )xx O=ψ
( ) ∑≤
=
primep
xp
px ln:θ
Theorem
Proof (continue):
Definition of O:
( ) ( ) ( ) ( ) ( ) +++==Λ= ∑∑
≤≤
3/2/ln: xxxpnx
primep
xpxn k
θθθψ
For x → ∞ and δ→0 we have
For 0 < δ < 1 and y = x1-δ
, we have
( ) ( ) ( ) ( )
( )2ln12
1
1ln
1
1ln 2ln12
+⋅
−
+≤⋅
−
+≤
+≤
δ
θ
δ
ψ
δ
θ
δ
x
x
x
x
x
x
x
x xx
0
ln
→δ
x
x
( ) ( ) xx
x
2ln12 +≤
∞→
ψ
Therefore ( ) ( )xx O=ψ
q.e.d.
Return to TOC

47
SOLO Primes
Riemann's Zeta Function (1859)
The Riemann Zeta Function or Euler–Riemann Zeta Function,
ζ(s), is a function of a complex variable s that analytically
continues the sum of the infinite series
( ) tis
n
s
n
s
+== ∑
∞
=
σς
1
1
“On the Number of Primes Less Than a Given Magnitude”, 7 page
paper offered to the Monatsberichte der Berliner Akademie on
October 19, 1859. The exact publication date is unknown.
( ) ( ) ( )s
s
ss ss
−





−Γ= −
1
2
sin12 1
ς
π
πς
where Γ(s) is the Gamma Function, which is an equality of
Meromorphic Functions valid on the whole complex plane. This
equation relates values of the Riemann Zeta Function at the points
s and 1 − s. The functional equation (owing to the properties of sin )
implies that ζ(s) has a simple zero at each even negative integer
s = −2n — these are known as the trivial zeros of ζ(s). For s an even
positive integer, the product sin(πs/2)Γ(1−s) is Regular and the
functional equation relates the values of the Riemann Zeta Function
at odd negative integers and even positive integers.
Georg Friedrich Bernhard Riemann
)1826–1866(
Return to TOC
To construct the analytic Continuation of the Zeta Function,
Riemann established the relation (see proof ).

Graph showing the Trivial Zeros, the
Critical Strip and the Critical Line
of ζ (s) zeros.
SOLO Primes
( ) ( )
( )
,2,1
1
1 1
=
+
−−=− +
n
n
B
n nn
ς
Those roots are called the Trivial Zeros
of the Zeta Function. The remaining
zeros of ζ (s) are called Nontrivial Zeros
or Critical Roots of the Zeta Function.
The Nontrivial Zeros are located on a
Critical Strip defined by 0 < σ < 1.
Since Bn+1 = 0 for n + 1 odd (n even)
we also have ( ) ,2,102 ==− mmς
We found
( ) { } σσς =+=
−
== ∏∑ −
∞
=
tis
pn
s
primep
z
n
s
Re
1
11
1
Riemann Zeta Function Zeros
Since the product contains no zero factors
we see that ζ (z) ≠ 0 for Re {z} >1.
Riemann Conjecture in his paper was
that all Zeta Function Nontrivial Zeros
are located at σ = ½. This Conjecture was
not proved and is named One of the
Greatest Mysteries in Mathematics.
Bn are the Bernoulli numbers

49
SOLO Primes
Riemann's Zeta Function
Specific Values
( ) ( ) ( )
( )
,3,2,1,0
!22
2
12
2
21
=−=
+
n
n
B
n
n
nn π
ς
For any positive even number 2n
where B2n are the Bernoulli numbers.
( ) ( ) ,3,2,1
1
1 1
=
+
−−=− +
n
n
B
n nn
ςFor negative integers one has
Therefore ζ vanishes at the negative even integers ζ (-2m) = 0 since B2m+1 = 0 for all m ,
m=1,2,…
( ) ,3,2,1
2
1
21 2 ==− mB
m
m mς
It is easy to show that the last equation is equivalent with
( ) ( )
2
1
2
10
1
01
0
−=−=
=B
B
ς

50
SOLO Primes
The Riemann zeta function or Euler–Riemann zeta function,
ζ(s), is a function of a complex variable s that analytically
continues the sum of the infinite series
( ) tis
n
s
n
s
+== ∑
∞
=
σς
1
1
which converges when the real part of s is greater than 1.
More general representations of ζ(s) for all s are given
below. The Riemann zeta function plays a pivotal role in
analytic number theory and has applications in physics,
probability theory, and applied statistics.
Georg Friedrich Bernhard
Riemann
1826 - 1866

51
SOLO Primes
Riemann's Zeta Function ( ) tis
n
s
n
s
+== ∑
∞
=
σς
1
1
Georg Friedrich Bernhard Riemann
)1826–1866(
Riemann zeta function ζ(s) in
the complex plane. The color
of a point s encodes the value
of ζ(s): colors close to black
denote values close to zero,
while hue encodes the value's
argument. The white spot at
s = 1 is the pole of the zeta
function; the black spots on
the negative real axis and on
the critical line Re(s) = 1/2 are
its zeros. Values with
arguments close to zero
including positive reals on the
real half-line are presented in
red

52
SOLO Primes
Riemann imaginary landscape
Graph showing the Trivial Zeros, the
Critical Strip and the Gritical Line
of ζ (s) zeros.
Modulus |ζ s)| ploted over the complex plane

53
SOLO Primes
The plots above show the real and imaginary parts of plotted in the complex plane together with
the complex modulus of ζ (s) . As can be seen, in right half-plane, the function is fairly flat, but
with a large number of horizontal ridges. It is precisely along these ridges that the nontrivial zeros
of ζ (s) lie.

54
Primes

55
Re ζ (s) in the original domain, Re s > 1.
Re ζ (s) after Riemann’s extension.
Primes

56
SOLO Primes
The position of the complex zeros can be seen
slightly more easily by plotting the contours of
zero real (red) and imaginary (blue) parts, as
illustrated above. The zeros (indicated as
black dots) occur where the curves intersect
The figures bellow highlight the zeros in
the complex plane by plotting |ζ(s)|) where
the zeros are dips) and 1/|ζ(s)) where the
zeros are peaks).

57
The Riemann Hypothesis
The Non-Trivial Zeros ρ of ζ (s) has Re ρ = 1/2
Primes

58
SOLO Primes
Year Number of zeros Computed by
1859 (approx.) 1 B. Riemann
1903 15 J. P. Gram
1914 79 R. J. Backlund
1925 138 J. I. Hutchinson
1935 1,041 E. C. Titchmarsh
1953 1,104 A. M. Turing
1956 15,000 D. H. Lehmer
1956 25,000 D. H. Lehmer
1958 35,337 N. A. Meller
1966 250,000 R. S. Lehman
1968 3,500,000 J. B. Rosser, et al.
1977 40,000,000 R. P. Brent
1979 81,000,001 R. P. Brent
1982 200,000,001 R. P. Brent, et al.
1983 300,000,001 J. van de Lune, H. J. J. te Riele
1986 1,500,000,001 J. van de Lune, et al.
2001 10,000,000,000 J. van de Lune (unpublished)
2004 900,000,000,000 S. Wedeniwski
2004 10,000,000,000,000 X. Gourdon
Computation of the Non-trivial Zeros of the Riemann Zeta Function.
All were on the Critical Line σ = ½.
Riemann Conjecture in his
paper was that all Zeta
Function Nontrivial Zeros
are located at σ = ½. This
Conjecture was not proved
and is named One of the
Greatest Mysteries in
Mathematics.
Return to TOC

59
SOLO Primes
( )
( )
( )∫
∞
−−
=
−
1
1'
uduu
ss
s s
ψ
ς
ς
We found
( ) ( )
( )∫
∞+
∞−
−
=
ic
ic
s
sd
s
x
s
s
i
x
ς
ς
π
ψ
'
2
1
Mellin Transform
( ) ( )
( ) 1,
1
ln
2
>+=
−
= ∫
∞
σσ
π
ς tisxd
xx
x
ss s
( ) ∑≤
=≤=
primep
xp
( )
( )( )
( ) ( ) ( )
( )
∏
<<
=
−−






−
+Γ−
=
10
0
2/12ln
1
2/112
ρ
ρς
ρ
γπ
ρ
ς
Re
s
e
s
ss
e
s
Hadamard
γ is the Euler-Mascheroni constant
γ=0.57721566490153286060651
( ) ( ) ( ) +−+−++
−
=
2
210 11
1
1
ss
s
s γγγς
( ) ( )






+
−
−
=
+
≤
∞→
∑ 1
lnln
lim
!
1 1
k
N
m
m
k
k
Nm
k
N
k
kγ
( ) ( ) ( ) ( ) ( ) +++==Λ= ∑∑
≤≤
3/2/ln: xxxpnx
primep
xpxn k
θθθψ
( )


 >=
=Λ
otherwise
n
k
0
1895
( ) ∑≤
=
primep
xp
px ln:θ

60
SOLO Primes
We found
( ) [ ]( ) 1
1
1
1
1
>−−=
−
− ∫
∞
−−
σ
ς
xdxxx
ss
s s
( ) ( ) 1
1
1lim
1 1
11 =
−
−=
−
=
→
==
s
s
s
s
s
s
s
ss ResRes ς
( )
( )




−
=
=
primesdistinctkofproducttheisnif
factorprimemultiplesomecontainsnif
nif
n
FunctionbiusoM
k
1
0
11
µ

( )



>
=
=∑ 10
11
| nif
nif
d
nd
µ
( )
( )
∑
∞
=
=
1
1
n
s
n
n
s
µ
ς
( ) [ ] 1
1
1
>= ∫
∞
−−
σς xdxxss s
Mellin Transform
[ ] ( )
∫
∞+
∞−
− −
−=
ic
ic
s
sd
s
s
x
i
x
ς
π2
1

61
SOLO Primes
We found
( ) ( ) ( )
( )
( )
( ) ( ) ∑
∑
∞
=
∞
=








=+







+







+







+=





−
=
=
=
1
1
4
1
3
1
2
1
1
/1
1
4
1
3
1
2
1
:
1
0
11
n
n
k
n
n
x
n
xxxxxJ
FunctionbiusoM
nif
n
xJ
n
n
x
πππππ
µ
µ
π


( ) ( )
( ) 1,
1
ln
2
>+=
−
= ∫
∞
σσ
πς
tisxd
xx
x
s
s
s
( ) ∑≤
=≤=
primep
xp
( ) ∑
∞
=








=
1
1
1
n
n
x
n
xJ π
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( )
∫
∫
∫∫
∑
∞+
∞−
−
∞
−
∞
−−
∞
−−
∞
=
−
−=
=
−
−
==
=⇔=≤≤







=+







+







+







+=
ic
ic
s
s
ss
n
n
sd
s
s
x
i
xJ
xdxxJ
s
s
xdxxJxdxxJ
s
s
Jxxx
n
xxxxxJ
ς
π
ς
ς
ππππππ
ln
2
1
ln
ln
00010
1
4
1
3
1
2
1
:
0
1
0
1
1
1
1
1
4
1
3
1
2
1

( )
( )
( )[ ] ( ) ∑∫∫
−
∞
−∞−
=
=
∞
−−
=+−=
−−
p
sss
xdxdv
xu
s
pxdxxxxdxxs
s
10
1
1
1
1
πππ
π


62
SOLO Primes
We found
( ) ( ) ( )
( ) ∫
∑
=
=
∞
=
x
n
n
x
dx
xLi
xLi
n
n
xR
0
1
/1
ln
:
:
µ
( ) ( )
∑
∞
=
++=
1 !
ln
lnln
n
n
nn
x
xxLi γ
( ) ( ) ( )
( ) ( )
( )
( ) ( )
( )∑∑ ∑
∞
=
∑ =
+=
∞
=
∞
=
+
+
+=+=
∞
=
1
1
1
1 1
1
1!
1
!
1
1
m
mmn
n
nn
m n
m
m
mmm
t
n
n
mm
t
xR
n
m
ς
µ ς
µ
µµ

63
SOLO Primes
( ) ( ) ( )xxLix lnππ O+=
( ) ∫=
x
t
td
xLi
2
ln
:
( ) ∑≤
=≤=
primep
xp
( ) ( )xxx 2
lnπψ O+=
( ) ( )nnLipn
2/51
lnπO+= −
( ) Constant
ln
1
1 2/1
Eulerx
x
e
pxp
−+=





− −
−
≤
∏ γ
γ
O
( )
( )( )
( ) ( ) ( )
( )
∏
>
=
−−






−
+Γ−
=
0Im
0
2/12ln
1
2/112
ρ
ρς
ρ
γπ
ρ
ς
s
e
s
ss
e
s Hadamard
Definition of O:
We say that f (x) = O (g (x)) if exists a constant k > 0 such that
|f (x)| < k |g (x)| Return to TOC

64
SOLO Primes
Riemann Zeta Function
( ) 


>
<
=∫ =
11
10
2
1
2:Re
aif
aif
ds
s
a
i ss
s
π
Special case of Perron’s Formula
Chebychev Psi Function
( ) ( )
( ) ( )
∫ ∑∑ ∫∑
=
≥
≤
≥
≤ =
=
≥
≤
===
2:Re
11
2:Re
/
1
ln
2
1/
ln
2
1
ln:
ss
s
m
primep
xp
ms
m
primep
xp ss
smPerron
xa
m
primep
xp
ds
s
x
p
p
i
ds
s
px
p
i
px
mm
m
m ππ
ψ
ρ
We were able to swap the infinite sum and the infinite integral since the terms are
convergent as Re (s) = 2
( ) ( ) 1
1
1lnln
1
>+==−−= ∑ ∑∑
∞
=
−
σσς tis
pm
ps
primep m
ms
SeriesTaylor
primep
s
( ) ( )
( )
( ) 1
ln
1
ln1'
ln
1
1
1
>+=−=
−
−−
−== ∑ ∑ ∑≥
−
−
−
−
σσ
ς
ς
ς tis
p
p
p
pp
s
s
s
sd
d
primep primep m
ms
Taylor
p
s
s
s
( ) tis
pn
s
primep
s
n
s
+=





−== ∏∑
−
∞
=
σς
1
1
1
1
1
( )
( )
( )
( )( )
∫∫ ∑
==
≥
≤
−
==
2:Re2:Re
1
'
2
1ln
2
1
ss
s
ss
s
m
primep
xp
ms
ds
s
x
s
s
i
ds
s
x
p
p
i
x
m ς
ς
ππ
ψ
Von Mangoldt Psi
Formula
Hans Carl Friederich von
Mangoldt 1895
( ) ( )
( ) ( )
( )
( ) 2/12
0Re
0
1
1ln
0
0'
ln: −
>
=
≥
≤
−−−−== ∑∑ x
x
xpx
m
primep
xpm
ρ
ρς
ρ
ρς
ς
ψ

65
SOLO Primes
( )
( )
( )
( )( )
∫∫ ∑
==
≥
≤
−
==
2:Re2:Re
1
'
2
1ln
2
1
ss
s
ss
s
m
primep
xp
ms
ds
s
x
s
s
i
ds
s
x
p
p
i
x
m ς
ς
ππ
ψ
(continue – 1)
Therefore
Define a semi-circular path CL (left side),
with s=2 as the origin., and R → ∞.
( )
( )
( )
( )
( )
( )
( )
( )
0
''
sincos
''
0cos
0
cos
cos
sincos
,,
∞→
<
>
+
→
−
=
−
=
+
−
≤
−
∫∫
∫∫
R
x
C
R
C
R
C
i
iRR
C
s
LL
RLRL
dx
s
s
dR
R
x
s
s
deRi
iRR
x
s
s
ds
s
x
s
s
ϕ
ϕ
ϕ
ϕ
ϕϕ
ϕ
ς
ς
ϕ
ς
ς
ϕ
ϕϕς
ς
ς
ς
( ) ( )
( )( )
( )
( )( )
( )
( )
( )
( )( )
∫∫∫∫ +===
−
=
−
+
−
=
−
=
LL Cs
s
C
s
ss
s
ss
s
ds
s
x
s
s
i
ds
s
x
s
s
i
ds
s
x
s
s
i
ds
s
x
s
s
i
x
2Re
0
2:Re2:Re
'
2
1'
2
1'
2
1'
2
1
ς
ς
πς
ς
πς
ς
πς
ς
π
ψ
  
( )
( )
( )
( )
( )
( ) ( )
( )
( ) 




−
+




−
+




−
=




−
=
→→+= s
x
s
s
s
x
s
s
s
x
s
s
s
x
s
s s
sofzeros
s
s
s
s
s
Cs L ς
ς
ς
ς
ς
ς
ς
ς
ς
'
Residues
'
Residue
'
Residue
'
Residues
102)Re(
( ) ( )
( ) ( )
( )
( ) 2/12
0Re
0
1
1ln
0
0'
ln: −
>
=
≥
≤
−−−−== ∑∑ x
x
xpx
m
primep
xpm
ρ
ρς
ρ
ρς
ς
ψ

66
SOLO Primes
(continue – 2)
( ) ( )
( )( )
∫ +=
−
=
LCs
s
ds
s
x
s
s
i
x
2Re
'
2
1
ς
ς
π
ψ
( )
( )
( )
( ) ( )
( )
( )
( )
( )
( ) ( )
( )
( ) ( )
( )( )
( ) ( )
( )( )
{ }
    
1Re0
0
0
,...4,2
0
0
1
1
0
0
10
'
lim
'
lim
1
'
1lim
'
lim
'
Residues
'
Residue
'
Residue
<<−
>
=
→
−−=
<
=
→→→
→→
∑∑ 




 −
⋅−+




 −
⋅−+




 −
⋅−+




 −
⋅=





−
+




−
+




−
=
ρ
ρ
ρς
ρ
ρ
ρ
ρ
ρς
ρ
ρ
ς
ρς
ς
ρ
ρς
ς
ρ
ς
ς
ς
ς
ς
ς
ς
ς
ς
ς
ZerosTrivialNon
s
ZerosTrivial
sss
s
sofzeros
s
s
s
s
x
s
s
s
x
s
s
s
x
s
s
s
s
x
s
s
s
s
x
s
s
s
x
s
s
s
x
s
s
( )
( )
( )
( )
( ) ( ) ( ) ( )
2
1
0&2ln
2
1
0'2ln
0
0''
lim
0
0
−=−=↔−=−=




 −
⋅
→
ςπςπ
ς
ς
ς
ς
s
x
s
s
s
s
( ) ( )
( ) ( )
( )
( ) 2/12
0Re
0
1
1ln
0
0'
ln: −
>
=
≥
≤
−−−−== ∑∑ x
x
xpx
m
primep
xpm
ρ
ρς
ρ
ρς
ς
ψ
( ) ( )
( ) ( )
( )( ) xx
s
sx
s
s
s
ss
=−




 −
=




 −
⋅−
→→ 
 1
1
1
1
1
1'
1
lim
1
'
1lim ς
ςς
ς
Now we have:

67
SOLO Primes
(continue – 3)
( ) ( )
( )( )
∫ +=
−
=
LCs
s
ds
s
x
s
s
i
x
2Re
'
2
1
ς
ς
π
ψ
( )
( )
( ) ( )
( )
( ) ( )
( )( )
( ) ( )
( )( )
{ }
    
1Re0
0
0
,...4,2
0
0
1
1
0
0
'
lim
'
lim
1
'
1lim
'
lim
<<−
>
=
→
−−=
<
=
→→→
∑∑ 




 −
⋅−+




 −
⋅−+




 −
⋅−+




 −
⋅=
ρ
ρ
ρς
ρ
ρ
ρ
ρ
ρς
ρ
ρ ρς
ς
ρ
ρς
ς
ρ
ς
ς
ς
ς
ZerosTrivialNon
s
ZerosTrivial
sss
x
s
s
s
x
s
s
s
x
s
s
s
s
x
s
s
s
( ) ( )
( )( ) ( )
( )
( )
( )( ) ( ) 2/12
1
2
1
2
1
0
0
2
,4,2
0
0
1ln
22
2'
2
lim
'
lim −
←∞
=
−∞
=
−
−
<
=
−→
−−=
<
=
→
−−=





−
−=
−
⋅





−−⋅
+
=




 −
⋅− ∑∑∑ x
n
x
n
x
n
s
nsx
s
s
s
Taylor
n
n
n
n
ns
ZerosTrivial
s
    

ς
ςρς
ς
ρ
ρ
ρς
ρ
ρ
ρς
ρ
ρ
( ) ( )
( )
( )
( ) ( )
( )( ) ( )
∑∑∑
>
=
>
=
−
→
−
=
>
→
−=




 −
⋅−=




 −
⋅−
0
0
0
0
1
0
0
'
lim
'
lim
ρ
ρς
ρ
ρ
ρς
ρ
ρ
ρς
ρ
ρ
ρ ρρς
ς
ρ
ρς
ς
ρ
xx
s
s
s
x
s
s
s
s
ZerosTrivialNon
s
  
  
( ) ( )
( ) ( )
( )
( ) 2/12
0Re
0
1
1ln
0
0'
ln: −
>
=
≥
≤
−−−−== ∑∑ x
x
xpx
m
primep
xpm
ρ
ρς
ρ
ρς
ς
ψ
q.e.d.
( )
( )
( )
( )( )
( )
( )( ) 1'
'
1
'lim
'
0
−=−⋅=−




 −
=
→
ρς
ρς
ρς
ς
ρ
ρς
ρ
HopitalL
s s
s
We also have:

68
SOLO Primes
Von Mangoldt showed that ψ can also be determined from the non-trivial zeros
ρ of the Zeta Function ζ (ρ) = 0
( )
( )
πρ
ψ
ρς
ρ
ρ
ρ
2
2/1
lnln
2
0
10
1
xx
xpx
m
xp
primep
m
−
−−== ∑∑
=
<<
≥
≤
Because the zeros ρ are complex, the values xρ
/ρ are also complex.
But since the nontrivial zeros come in complex-conjugate pairs ρ
and ρ*. The values xρ
/ρ and xρ*
/ρ* are also complex conjugate so
all imaginary parts cancel in the infinite sum.
The function xρ
/ρ maps the positive reals onto a logarithmic spiral in the complex
plane. xρ
/ρ and xρ*
/ρ* produce complex conjugate spirals (mutual reflections across the
real axis. xρ
/ρ + xρ*
/ρ* =2 Re [xρ
/ρ] is a real valued function, a sort of logarithmically –
rescaled sinusoid with increased amplitude as pictured bellow:
...)13.14(2/1 i+=ρ ...)58.37(2/1 i+=ρ
Von Mangoldt Psi Formula (continue – 4)

69
SOLO Primes
( )
( )
πρ
ψ
ρς
ρ
ρ
ρ
2
2/1
lnln
2
0
10
1
xx
xpx
m
xp
primep
m
−
−−== ∑∑
=
<<
≥
≤
Comparing ψ (x) with its approximation via
summing the first 50 zeros of the Zeta function.
The Chebyshev Psi Function can be reconstructed by starting with the function
x – ln (2π)-1/2 ln (1-1/x2
), and then successively adding “spiral wave” functions.

70
SOLO Primes
( )
( )
πρ
ψ
ρς
ρ
ρ
ρ
2
2/1
lnln
2
0
10
1
xx
xpx
m
xp
primep
m
−
−−== ∑∑
=
<<
≥
≤
Comparing ψ (x) in the interval x (2.5,ϵ
5.5) with its approximation via summing the
first 100 zeros of the Zeta function.
Comparing ψ (x) in the interval x (2.5, 5.5)ϵ
with its approximation via summing the first
500 zeros of the Zeta function.
The Chebyshev Psi Function can be reconstructed by starting with the function
x – ln (2π)-1/2 ln (1-1/x2
), and then successively adding “spiral wave” functions.
( )
( )
πρ
ψ
ρς
ρ
ρ
ρ
2
2/1
lnln
2
0
10
1
xx
xpx
m
xp
primep
m
−
−−== ∑∑
=
<<
≥
≤

71
SOLO Primes
( )
( )
πρ
ψ
ρς
ρ
ρ
ρ
2
2/1
lnln
2
0
10
1
xx
xpx
m
xp
primep
m
−
−−== ∑∑
=
<<
≥
≤
Let take the derivative of the staircase function ψ (x)
( ) ( )
( )
2/12
1
1' 2
0
10
1
x
x
xxx
xd
d
−
+−== ∑
=
<<
−
ρς
ρ
ρ
ρ
ψψ
Since ψ (x) is a staircase function that jumps at each prime power pk
, ψ’(x)
should be zero except for spikes at
2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19,…
In the sum
each conjugate pair contributes a waveform (harmonic mode)
( )
( )
2
1' 2
0
10
1
−
−−= ∑
=
<<
−
x
x
xx
ρς
ρ
ρ
ρ
ψ
{ }ρρ,
( ) ( ) ( )
( ) ( )
( )( )xxeexxx xixi
ln1cos2 1ln1ln1111
−=+=+ −−−−−−−
ρρρρρρρ
ImReImImRe
Since 0 < Re ρ < 1, we have -1 < Re (ρ-1) < 0, therefore the amplitude of the
waveform is a monotonic decreasing function of x. The frequency of
the waveform is related to Im (ρ – 1) ln x is a monotonic increasing function of x.
( )1
2 −ρRe
x

72
SOLO Primes
The effect of Riemann's harmonics
Riemann's harmonics

73
SOLO Primes
For example here are plots of ψ’(x) using
Nρ=10, 50 and 200 pairs of zeros
ψ’(x) is zero except for spikes at
2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19,…
Nρ = 10
Nρ = 50
Nρ = 200
( ) ( )
( )
2/12
1
1' 2
0
10
1
x
x
xxx
xd
d
−
+−== ∑
=
<<
−
ρς
ρ
ρ
ρ
ψψ

74
SOLO Primes
Each conjugate pair contributes a waveform (harmonic mode){ }ρρ,
( )
( )( )xxxx ln1cos2 111
−=+ −−−
ρρρρ
ImRe
If the Riemann Hypothesis (R.H. = Re ρ = ½) is true all the harmonics will
have the same amplitude xx /22 2/1
=−
If the Riemann Hypothesis is not , that at least one harmonics has a
different amplitude then others.

75
SOLO Primes
( )
0
1
lim
0
10
=∑
=
<<
∞→
ρς
ρ
ρ
ρ
ρ
x
xx
But independent if the assumption that Riemann Hypothesis is true or false,
since we have 0 < Re ρ < 1 for all ρ, we have
From the Explicit Formula for ψ (x)
( )
( )
πρ
ψ
ρς
ρ
ρ
ρ
2
2/1
ln
11
1
2
0
10
x
x
x
xx
x −
−−= ∑
=
<<
Also 0
2
2/1
ln
1
lim
2
=
−
∞→ π
x
xx
Therefore that proves the Prime Number Theorem.
( ) 1lim =
∞→ x
x
x
ψ
Return to TOC

SOLO Primes
( ) ( )
( ) ( ) ∑
∫
∞
=
∞
−−








=+







+







+







+=
=
1
1
4
1
3
1
2
1
1
1
1
4
1
3
1
2
1
:
ln
n
n
s
x
n
xxxxxJ
xdxxJ
s
s
πππππ
ς

This sum is only formally infinite, since , as soon as decreases
bellow 2, which will happen as soon as n > lnx/ln2. f (x) has jumps of 1/r when x
passes a prime power pr
. (when x passes a prime p, this is regarded as the prime
power p1
.)
( ) 0/1
=n
xπ n
x /1
Proof:
( ) ( ) ( )
( )
+++=
−−=−=
∑∑∑
∑∏
≤
−
≤
−
≤
−
+
≤
−
≤
−−
primep
xp
s
primep
xp
s
primep
xp
s
a
Series
Taylor
primep
xp
s
primep
xp
s
ppp
pps
32
1ln
1
3
1
2
1
1ln1lnlnς

SOLO Primes
( ) ( )
( ) ( ) ∑
∫
∞
=
∞
−−








=+







+







+







+=
=
1
1
4
1
3
1
2
1
1
1
1
4
1
3
1
2
1
:
ln
n
n
s
x
n
xxxxxJ
xdxxJ
s
s
πππππ
ς

Proof (continue – 1):
( ) +++= ∑∑∑ ≤
−
≤
−
≤
−
primep
xp
s
primep
xp
s
primep
xp
s
ppps 32
3
1
2
1
lnς
Using Stieltjes’ Integrals and performing Integration by Parts, we obtain
( )
( )
( )[ ] ( ) ∑∫∫
−
∞
−∞−
=
=
∞
−−
=+−=
−−
p
sss
xdxdv
xu
s
pxdxxxxdxxs
s
10
1
1
1
1
πππ
π

This follows since and d π (x) will increase by 1
when x is a prime number p, and will be zero between primes.
( ) ( ) 0lim00
0
0
== −
∞→
−
xxx s
x
ππ
In the same way
∑∫∫
−
∞
−
∞
−
=








=
∞
−−
=







+
















−=






 −−
p
snnsns
xdxdv
xu
sn
pxdxxxxdxxs
s
n
1
1
0
1
1
1
1
1 1
1
πππ
π
  

SOLO Primes
( ) ( )
( ) ( ) ∑
∫
∞
=
∞
−−








=+







+







+







+=
=
1
1
4
1
3
1
2
1
1
1
1
4
1
3
1
2
1
:
ln
n
n
s
x
n
xxxxxJ
xdxxJ
s
s
πππππ
ς

Proof (continue – 2):
( )
( )
( )∫
∫∫
∑∑∑
∞
−−
∞
−−
∞
−−
≤
−
≤
−
≤
−
=
+







+=
+++=
1
1
1
12
1
1
1
32
2
1
3
1
2
1
ln
xdxxJs
xdxxsxdxxs
ppps
s
ss
primep
xp
s
primep
xp
s
primep
xp
s


ππ
ς
( ) ∑
∞
=








=
1
1
1
:
n
n
x
n
xJ π

SOLO Primes
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( )
∫
∫
∫∫
∑
∞+
∞−
−
∞
−
∞
−−
∞
−−
∞
=
−
−=
=
−
−
==
=⇔=≤≤







=+







+







+







+=
ic
ic
s
s
ss
n
n
sd
s
s
x
i
xJ
xdxxJ
s
s
xdxxJxdxxJ
s
s
Jxxx
n
xxxxxJ
ς
π
ς
ς
ππππππ
ln
2
1
ln
ln
00010
1
4
1
3
1
2
1
:
0
1
0
1
1
1
1
1
4
1
3
1
2
1

( ) ( )
( ) 1,
1
ln
2
>+=
−
= ∫
∞
σσ
πς
tisxd
xx
x
s
s
s
We found the following expressions for ln ζ(s)/s:
( ){ } ( ) ( )∫
∞
−
==
0
1
xdxfxsFxf s
MM
( ){ } ( ) ( )∫
∞+
∞−
−
==
ic
ic
s
sdsFx
i
x M
1-
fsfM
π2
1
Return to TOC

SOLO Primes
Abel’s Method of Partial Summation:
∑ ∑∑∑ ∑∑
+
= =
+
−
== ==
+





−





=
1
2 1
1
1
11 11
N
n
N
n
nN
n
i
in
N
n
n
i
in
N
n
nn babababa
∑ ∑∑∑ ∑ = =
+
=
+
= =
+





−





=
N
n
N
n
nN
n
i
in
N
n
n
i
in bababa
1 1
1
1
1
1 1
( )∑ ∑∑ = =
+
=
+ 





−−=
N
n
n
i
inn
N
n
nN baaba
1 1
1
1
1
( )( ) ∑∑∑ ==
++
=
=−−=
n
i
in
N
n
nnnNN
N
n
nn bBBaaBaba
11
11
1
:
1+
↓
n
n
Niels Henrik Abel
( 1802 – 1829)

SOLO Primes
Use Abel’s Method of Partial Summation:
( )( ) ∑∑∑ ==
++
=
=−−=
n
i
in
N
n
nnnNN
N
n
nn bBBaaBaba
11
11
1
:
( ) 1lim
1
>= ∑=
−
∞→
σς
N
n
s
N
nsfor:
by choosing an = n-s
, bn = 1, therefore Bn = n
( ) ( ) ( )( ) ( )( )∑∑∑
∞
=
−−
=
−−
∞→
−
∞→
=
−
∞→
+−⋅=+−⋅++==
11
0
1
11lim1limlim
n
ss
N
n
ss
N
s
N
N
n
s
N
nnnnnnNNns
  
ς
[ ]
[ ] xdxxsxdxns s
nx
n
n
n
s
∫∑ ∫
∞
−−
=∞
=
+
−−
=⋅=
1
1
1
1
1
Where [x] is the integer, less then x and closer to x
[ ] [ ] 10s.t.integer <−≤ xxx
( ) [ ] 1
1
1
>= ∫
∞
−−
σς xdxxss s
( ) [ ] [ ] [ ] 1
1 1 1
1
1
1 11 11
1
>
−
−
−
=
−
−== ∫∫∫∫
∞
+
∞+−
∞
+
∞
+
∞
−−
σς xd
x
xx
s
s
xs
xd
x
xx
sxd
x
x
sxdxxss s
s
ss
s
( ) [ ]( ) 1
1 1
1
>−−=
−
− ∫
∞
−−
σς xdxxxs
s
s
s s
Return to TOC

SOLO Primes
( ) [ ]( ) 1
1 1 1
>
−
−=
−
− ∫
∞
+
σς xd
x
xx
s
s
s
s s
( ) [ ] [ ]



=
<≤
=== ∫∑
∞
−=
−
∞→ 11
100
lim
11 xd
xd
xd
x
xd
ns s
N
n
s
N
ε
ςProof:
Integrating by parts:
( ) [ ]
[ ]
[ ] [ ] [ ] [ ]
[ ]( ) [ ]( )
∫∫
∫∫∫∫
∞
+
∞
+
∞+−
∞
+
∞
+
∞
−
+
∞
−
==
=−=
∞
−
−
−
−
−=
−
−
−
=
−
−=+==
−
−−
1
1
1
1
1
1
1
1
1
1
1
1
0
1
,
,
1
11
1
ss
s
ssss
xddvxu
xvdxxsdu
s
x
xdxx
s
s
s
x
xdxx
s
s
xs
x
dxxx
s
x
dxx
s
x
dxx
s
x
x
x
xd
s
s
s
εεε
ς

[ ]( ) 1
1
1
1 1
>≤
−
∑∫
∞
=
∞
+
σforconverges
n
xd
x
xx
n
ss
We can see that
We have an Analytic Continuation for by removing the singularity at s = 1
of ζ (s). We can see that ζ (s) can a simple pole at s=1, and
( )
1−
−
s
s
sς
( ) ( ) 1
1
1lim
1 1
11 =
−
−=
−
=
→
==
s
s
s
s
s
s
s
ss ResRes ς

SOLO Primes
( ){ } ( ) ( )∫
∞
−
==
0
1
xdxfxsFxf s
MM
( ){ } ( ) ( )∫
∞+
∞−
−
==
ic
ic
s
sdsFx
i
x M
1-
fsfM
π2
1
Mellin Transform
Inverse Mellin Transform
( ) ( ) [ ]∫
∞
−
=
−
−=−
0
1
xdxx
s
s
sF sς
M
[ ] ( )
∫
∞+
∞−
− −
−=
ic
ic
s
sd
s
s
x
i
x
ς
π2
1
( ) [ ] 1
1
1
>= ∫
∞
−−
σς xdxxss s
Return to TOC

SOLO Primes
( )
( )




−
=
=
nif
n
k
1
0
11
µ
Möbius Function
The most important property of Möbius function is
( )



>
=
=∑ 10
11
| nif
nif
d
nd
µ
The symbol d|n means that the integer d divides the integer n, therefore the sum is on
all integers d that divide n. (note that the improper divisor d=1 and d=n have to be
included in this formula)
To prove this property, suppose that with all pi being different primes.
Then d|n, and μ (d) = (-1)k
if d is a product of precisely k different members of the set
of s primes pi. This case will occur for different divisors d of n. All divisors d of n
containing one or several of the primes pi twice or more have μ (d) = 0, according to
the definition of μ (d). Thus
is
i ipn
α
∏=
= 1






k
s
( ) ( ) ( ) 1,0111
0|
≥=−=





−= ∑∑ =
sif
k
s
d
s
s
k
k
nd
µ
August Ferdinand Möbius
1790 - 1868
( ) ( ) 11
1|
==∑=
µµ
nd
d

SOLO Primes
( )
( )




−
=
=
nif
n
k
1
0
11
µ
Möbius Function
The most important property of Möbius function is
( )



>
=
=∑ 10
11
| nif
nif
d
nd
µ
Theorem: This relation has as one of its consequence that:
( )
( )
∑
∞
=
=
1
1
n
s
n
n
s
µ
ς
since:
( ) ( ) ( ) ( )
( )
( )
111
1
1
||
1 11
=⋅====⋅ −
∞
=
∞
=
∞
=
∞
=
∑
∑
∑
∑
∑ ∑∑ s
n
s
nd
s
mdd
m d
ss
n
s
n
d
dm
d
d
d
mn
n
s
µµµµ
ς
q.e.d.
Return to TOC

SOLO Primes
( ) ( ) ( )
( )
( )
( ) ( ) ∑
∑
∞
=
∞
=








=+







+







+







+=





−
=
=
=
1
1
4
1
3
1
2
1
1
/1
1
4
1
3
1
2
1
:
1
0
11
n
n
k
n
n
x
n
xxxxxJ
nif
n
xJ
n
n
x
πππππ
µ
µ
π

Proof
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )xd
u
x
x
u
m
u
x
nm
m
nm
m
x
n
n
xJ
n
n
n ud
u
n um
u
n m
mn
n m
mn
n
n
πµ
π
π
µ
π
µ
πµµ
=







=






=






=





=
∑ ∑∑∑
∑∑∑ ∑∑
∞
=
∞∞
=
∞
∞
=
∞
=
∞
=
∞
=
∞
=
1 |
/1
1 |
/1
1 1
/1
1 1
/1
1
/1
( )



>
=
=∑ 10
11
| nif
nif
d
nd
µ
Conversion from J (x) back to π (x)
q.e.d.
Return to TOC

SOLO Primes
( ) ( ) ( )
( )
( )
( ) ( ) ∑
∑
∞
=
∞
=








=+







+







+







+=





−
=
=
=
1
1
4
1
3
1
2
1
1
/1
1
4
1
3
1
2
1
:
1
0
11
n
n
k
n
n
x
n
xxxxxJ
nif
n
xJ
n
n
x
πππππ
µ
µ
π

Riemann defined the following formula to approximate the π (x):
The Riemann Prime Number Formula
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ∫
∑
=
−+−−−=
=
∞
=
x
n
n
x
dx
xLi
xLixLixLixLixLi
xLi
n
n
xR
0
6/15/13/12/1
1
/1
ln
:
6
1
5
1
3
1
2
1
:

µ

SOLO Primes
The Riemann Prime Number Formula (continue – 1)
( ) ( ) ( )
( ) ∫
∑
=
=
∞
=
x
n
n
x
dx
xLi
xLi
n
n
xR
0
1
/1
ln
:
:
µ
We can see from the Table
that R (x) gives a better
approximation of the π (x)
then Li (x)

SOLO Primes
( ) ( ) ( )
( ) ∫
∑
=
=
∞
=
x
n
n
x
dx
xLi
xLi
n
n
xR
0
1
/1
ln
:
:
µ
( )
( )
  
γ−
∞
=
−∞→
∞
=∞−
∞
=
∞−
∞
=
−
∞−
=
=






+−+=





+=
===
∑∑∑
∫ ∑∫∫
11
ln
1
ln
0
1ln
0
!
lnlim
!
ln
lnln
!
ln
!ln
:
n
n
t
n
nx
n
n
x
n
neof
Series
Taylor
x tex
dtedx
x
nn
t
t
nn
x
x
nn
t
t
n
dtt
td
t
e
x
dx
xLi
tt
t
( ) ( )
∑
∞
=
++=
1 !
ln
lnln
n
n
nn
x
xxLi γ

SOLO Primes
( ) ( ) ( )
( ) ∫
∑
=
=
∞
=
x
n
n
x
dx
xLi
xLi
n
n
xR
0
1
/1
ln
:
:
µ
( ) ( )
∑
∞
=
++=
1 !
ln
lnln
n
n
nn
x
xxLi γ
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
∑∑∑∑
∑ ∑∑∑
∞
=
∞
=
+
∞
=
∞
=
∞
=
∞
=
∞
=
=∞
=
+−+=








++===
1 1
1
11
1 11
/
1
/1
!
ln
ln
!
/
ln:
n m
m
m
nn
n m
m
n
nt
ex
n
n
mmn
tn
n
nn
n
n
t
mm
nt
n
t
n
n
eLi
n
n
xLi
n
n
xR
t
µµµ
γ
γ
µµµ
( ) ( )
( )
( )
0
1
limlim
1
1
1
1
1
∞→
→
∞
=
→
∞
=
=== ∑∑
ς
ς
µµ
sn
n
n
n
s
n
ss
n
But
( ) ( ) ( )
( )
( )
( )
( )
( )
( )
( )
( )
1
1
1
1
1
lim
'
lim
1
limlim
1
lim
ln
lim
ln
2
2
1211
1
1
1
1
1
1
1
−=
+
−
+
−
−
==





−=−=






−==
→→→
∞
=
→
∞
=
→
∞
=
→
∞
=
∑
∑∑∑
so
s
so
s
s
s
ssd
d
n
n
sd
d
nsd
d
n
n
n
nn
n
nn
sss
n
ss
n
sss
n
ss
n
ς
ς
ς
µ
µµµ

SOLO Primes
( ) ( ) ( )
( ) ∫
∑
=
=
∞
=
x
n
n
x
dx
xLi
xLi
n
n
xR
0
1
/1
ln
:
:
µ
( ) ( )
∑
∞
=
++=
1 !
ln
lnln
n
n
nn
x
xxLi γ
( ) ( ) ( )
( ) ( )
( )
( ) ( )
( )∑∑ ∑
∞
=
∑ =
+=
∞
=
∞
=
+
+
+=+=
∞
=
1
1
1
1 1
1
1!
1
!
1
1
m
mmn
n
nn
m n
m
m
mmm
t
n
n
mm
t
xR
n
m
ς
µ ς
µ
µµ
Return to TOC

92
SOLO Primes
Theorem
( ) ( )∑∑ ≤≤
Λ==
xn
primep
xp
npx
k
ln:ψ
For x ≥ 2 we have
( )


 >=
=Λ
otherwise
andpprimesomeforpnifp
n
0
1integerln αα
( ) ( ) ( ) ( )2/1
2
2
lnln
xOtd
tt
t
x
x
x
x
++= ∫
ψψ
π
Proof
( ) ∑≤
=
primep
xp
px ln:θDefine
then
( )
( )
( )
( )
( ) ( )
x
x
x
x
p
t
p
td
tt
p
td
tt
p
td
tt
t
x
xp
xx
xpxp
x
ptp
xx
tp
x
ln
1
ln
ln
ln
ln
ln
ln
ln
ln
ln
ln/
2
2
2
2
2
2
θ
π
θ
πθ
−=−−





−=





−===
−
≤
−
≤≤≤≤
∑∑∑∑ ∫∫ ∑∫

Von Mangoldt Function
( ) ( ) ( )
∫+=
x
td
tt
t
xx
x
xx
x
2
2
ln
1
ln/
θθπ
Return to TOC

93
SOLO Primes
Hadamard Proof of the Prime Number Theorem (1896)
Hadamard paper on PNT used the Riemann Zeta Function ζ (s) for
which he proed some new properties.
His paper published in 1896 consists of two parts:
In the First Part he proved that the Zeta Function has no Zeros on
the line Re (s) = σ = 1. His proof is complicated, hence here we give
the F. Mertens method to prove this.
Jacques Salomon
Hadamard
(1865 –1963)

94
SOLO Primes
Hadamard Proof of the Prime Number Theorem (continue - 1
Start with the Riemann Zeta Function
( ) ( ) 1
1
1lnln
1
>+==−−= ∑ ∑∑
∞
=
−
σσς tis
pm
ps
primep m
ms
SeriesTaylor
primep
s
( ) ( )
( )
( ) 1
1
ln
1
ln1'
ln
1
1
1
>+=





−=
−
−−
−== ∑ ∑ ∑≥
−
−
−
−
σσ
ς
ς
ς tis
p
p
p
pp
s
s
s
sd
d
primep primep m
ms
Taylor
p
s
s
s
( ) tis
pn
s
primep
s
n
s
+=





−== ∏∑
−
∞
=
σς
1
1
1
1
1
( ) 1
32
1ln
1
32
<=−++++=−− ∑=
x
m
x
m
xxx
xx
m
m
mmSeriesTaylor

Where the last series counts the prime powers pm
, with the weight ln p, therefore
( )


 >=
=Λ
otherwise
n
k
0
1895
( ) ( )
( )
( ) 1
1
ln
'
ln
11
>+=
Λ
−=





−== ∑∑ ∑ ≥≥
σσ
ς
ς
ς tis
n
n
p
p
s
s
s
sd
d
n
s
primep m
ms
Jacques Salomon
Hadamard
(1865 –1963)

95
SOLO Primes
Hadamard Proof of the Prime Number Theorem (continue - 2)
( )


 >=
=Λ
otherwise
n
k
0
1895
( )
( )
( ) 1
'
1
>+=
Λ
−= ∑≥
σσ
ς
ς
tis
n
n
s
s
n
s Jacques Salomon
Hadamard
(1865 –1963)
( )
( )( )
( ) ( ) ( )
( )
∏
<<
=
−−






−
+Γ−
=
10
0
2/12ln
1
2/112
ρ
ρς
ρ
γπ
ρ
ς
Re
s
e
s
ss
e
s
Hadamard Product Representation of Riemann Zeta Function
Hadamard established the following form of the Mellin Inversion Formula
∫ ∑∑
∞+
∞−
∞
=<
=




 i
i
n
s
n
s
xn
n sd
n
a
s
x
in
x
a
2
2
1
2
2
1
ln
π
Substitute an = Λ (n)
( ) ( ) ( )
( )∫∫ ∑∑
∞+
∞−
∞+
∞−
∞
=<
−=
Λ
=





Λ
i
i
s
i
i
n
s
s
xn
sd
s
s
s
x
i
sd
n
n
s
x
in
x
n
2
2 2
2
2
1
2
'
2
1
2
1
ln
ς
ς
ππ

96
SOLO Primes
Hadamard Proof of the Prime Number Theorem (continue - 3)
Jacques Salomon
Hadamard
(1865 –1963)
( ) ( ) ( )
( )∫∫ ∑∑
∞+
∞−
∞+
∞−
∞
=<
−=
Λ
=





Λ
i
i
s
i
i
n
s
s
xn
sd
s
s
s
x
i
sd
n
n
s
x
in
x
n
2
2 2
2
2
1
2
'
2
1
2
1
ln
ς
ς
ππ
( )
( )
( ) { } { } 1
1
1
>=+==− ∫
∞
−−
σσψ
ς
ς
tisuduus
s
s
zd
d
s
ReRe
( ) ( )
( )∫
∞+
∞−
−
=
ic
ic
s
sd
s
x
s
s
i
x
ς
ς
π
ψ
'
2
1
( ) ( ) ∑∑
≤≤
=Λ=
primep
xpxn k
pnx ln:ψ
Return to TOC

97
SOLO Primes
Newman’s Proof of the Prime Number Theorem (1980)
Proofs have introduced various simplifications to Hadamard and
de la Vallée-Poussin through the use of Tauberian theorems but
remained difficult to digest, a surprisingly short proof was
discovered in 1980 by American mathematician Donald J.
Newman. Newman's proof is arguably the simplest known proof
of the theorem, although it is non-elementary in the sense that it
uses Cauchy's integral theorem from complex analysis
Donald J. Newman
( 1930 –2007)
Prime Number Theorem
( ) 1
ln/
lim =
∞→ xx
x
x
π
Newman’s Proof:
( ) ( ) ( )
x
x
x
x
xx
x
xxx
ψθπ
∞→∞→∞→
== limlim
ln/
limSince we proved that it is enough to prove that
( ) 1lim =
∞→ x
x
x
θ
Newman started by proving that
( ) 0
1
2
→
−
∫
∞
xd
x
xxθ
First suppose that exists λ > 1 such that θ (x) ≥ λ x for all x sufficiently large (say x ≥ x0)
( ) 0
1
0
2
1
2222
>
−
=
−
=
−
≥
−
∫∫∫∫
>
=
=
λλλλ
λλλθ
ud
u
u
udx
xu
xux
td
t
tx
td
t
tt x
t
u
x
td
ud
x
x
x
x

Now suppose that exists λ < 1 such that θ (x) ≤ λ x for all x sufficiently large (say x ≥ x0)
This is a contradiction to
( ) 0
1
2
→
−
∫
∞
xd
x
xxθ
( ) 0
1
0
2
1
2222
<
−
=
−
=
−
≤
−
∫∫∫∫
<
=
= λλλλ
λλλθ
ud
u
u
udx
xu
xux
td
t
tx
td
t
tt x
t
u
x
td
ud
x
x
x
x

This is a contradiction to
( ) 0
1
2
→
−
∫
∞
xd
x
xxθ
Therefore the only possibility is:
( ) 1lim ==
∞→
λ
θ
x
x
x

98
SOLO Primes
Newman’s Proof of the Prime Number Theorem
Donald J. Newman
( 1930 –2007)
Newman started by proving that
( ) 0
1
2
→
−
∫
∞
xd
x
xxθ
This is done in the following steps:
( ) ( ) ( ) ( ) ( )∫∫∫∑
∞
−
=
=
∞
+
∞=
=
=
−=
∞
=+===Φ
−
−−
01
1
0
11
1
ln
: dteezdx
x
x
z
x
x
x
xd
p
p
z tzt
ex
dtexd
zz
ddv
xu
v
dxzxdu
z
primep
p
z
t
t
z
z
θ
θθθ θ
θ 
Newman’s Proof (continue – 1):
Define:
Prove that:
( ) ( ) ( )( )∫∫∫
∞
−
∞=
=
∞
−=
−
=
−
00
2
1
2
1 tdeetde
e
ee
xd
x
xx ttt
t
ttex
tdexd
t
t
θ
θθ
( ) ( ) 1: −= −tt
eetf θ
( ) ( ) ( ) ( ) ( )
zz
z
tdetdeetdetfzF tztzttz 1
1
1
:
00
1
0
−
+
+Φ
=−== ∫∫∫
∞
−
∞
+−
∞
−
θ
( ) ( ) ???0
1
1
1
limlim
00
=





−
+
+Φ
=
+→+→ zz
z
zF
zz
Apply Analytical Theorem – A Tauberian Theorem
( ) ( ) ( ) 00
1
1
1
limlim
1
200
→
−
⇔=





−
+
+Φ
= ∫
∞⇓
+→+→
xd
x
xx
zz
z
zF
zz
θ
q.e.d.

99
SOLO Primes
Donald J. Newman
( 1930 –2007)
( ) ( ) ( ) ( ) ( )∫∫∫∑
∞
−
=
=
∞
+
∞=
=
=
−=
∞
=+===Φ
−
−−
01
1
0
11
1
ln
: dteezdx
x
x
z
x
x
x
xd
p
p
z tzt
ex
dtexd
zz
ddv
xu
v
dxzxdu
z
primep
p
z
t
t
z
z
θ
θθθ θ
θ 
Use the Identity:
( )
( )
( )
( ) 1
1
ln
1
ln1
ln >+=
−
−=
−
−−
−== ∑ ∑−
−
σσ
ς
ς
ς tiz
p
p
p
pp
z
z
zd
d
z
zd
d
primep primep
zz
z
( )1
11
1
1
−
+=
− zzzz
pppp
We found:
( )
( ) ( ) ( )
( ) 1
1
ln
1
lnln
1
ln
>+=
−
+Φ=
−
+=
−
=− ∑∑∑∑ σσ
ς
ς
tiz
pp
p
z
pp
p
p
p
p
p
z
z
zd
d
primep
zz
primep
zz
primep
z
primep
z
The sum is:
( ) ( ) ( ) 2/112
ln
1
ln
2
>⇔>≈
−
∑∑ zzforconvergent
p
p
pp
p
primep
z
primep
zz
ReRe

100
SOLO Primes
Donald J. Newman
( 1930 –2007)
We found:
( )
( ) ( ) ( )
( ) 1
1
ln
1
lnln
1
ln'
>+=
−
+Φ=
−
+=
−
=− ∑∑∑∑ σσ
ς
ς
tiz
pp
p
z
pp
p
p
p
p
p
z
z
primep
zz
primep
zz
primep
z
primep
z
( ) ( ) ( ) 2/112
ln
1
ln
2
>⇔>≈
−
∑∑ zzforconvergent
p
p
pp
p
primep
z
primep
zz
ReRe
Change z to z+1:
We found:
We proved also that:
( )
( )
( )
( ) 1
1
ln1
1
1
1
1'
≥
−
+
−
−
Φ
=
−
−− ∑ σ
ς
ς
foranalytic
pp
p
zzz
z
zzz
z
primep
zz
( )
( ) ( )
( )
( ) 0
1
ln
1
11
1
11
11
1'
≥
−+
+−
+
+Φ
=−
++
+
− ∑ σ
ς
ς
foranalytic
pp
p
zzz
z
zzz
z
primep
zz
( ) 0
1
1
1
lim
0
=





−
+
+Φ
→ zz
z
z
Stil need to prove

101
SOLO Primes
Donald J. Newman
( 1930 –2007)
Analytical Theorem – A Tauberian Theorem
Proof of the Analytic Theorem
( ) ( )∫
−
=
T
ts
T dtetfsF
0Consider the sequence of functions
Those functions are entire (analytic), and we are
trying to show that limT→∞ FT (0) exists and is
equal to F (0).
Let chose a closed counterclockwise path of
integration γR composed from a semicircle γR
+
(z)
{z C| |z|≤ R, Re(z)>-δϵ }, where we choose δ > 0
small enough (depending on R) so that
F (z) is analytic inside γR. (Such a δ exists by
compactness and the fact that F (z) is analytic
for Re (z) ≥ 0)
Let f (t) be a bounded and locally integrable function for
t ≥ 0, and suppose that when Re (z) ≥ 0
extends holomorphically to Re (z) ≥ 0. then exists
and equals F (0).
( )∫
∞
0
dttf
( ) ( )∫
∞
−
=
0
dtetfsF ts

102
SOLO Primes
Donald J. Newman
( 1930 –2007)
Proof of the Analytic Theorem (continue – 1)
Let use the Cauchy Theorem to compute
and equals F (0).
( )∫
∞
0
dttf
( ) ( )∫
∞
−
=
0
dtetfsF ts
The additional term z2
/R2
was introduced by Newman in order to help the proof.
( ) ( )( ) ( ) ( )( ) ( ) ( )00
1
1lim1
2
1
2
2
02
2
T
zT
T
z
Cauchy
zT
T FF
zR
z
ezFzFz
z
zd
R
z
ezFzF
i R
−=





+−⋅=





+−
→∫γπ
( ) ( )( )∫+






+−
R
z
zd
R
z
ezFzF
i
zT
T
γ
π 2
2
1
2
1Start with the integral on γR
+
( ) ( ) ( ) ( )
( )
( )z
eB
tdetftdetfzFzF
Tz
T
st
B
t
T
st
T
Re
max
Re
0
−
∞
−
≥
∞
−
=≤=− ∫∫ 
( ) ( ) ( ) ( ) ( )
2
Re
2
*
Re
2
22
Re
2
2
Re21
1
R
z
e
zR
zzz
e
zR
zR
e
zR
z
e TzTzTzzT
=
+
=
+
=





+
( ) ( )( )
( )
( )
( ) ( )
R
B
R
z
e
z
eBR
z
zd
R
z
ezFzF
i
Tz
Tz
zT
T
R
=⋅≤





+−
−
∫+
2
Re
Re
2
2
Re2
Re2
1
2
1
π
π
π γ

103
SOLO Primes
Donald J. Newman
( 1930 –2007)
and equals F (0).
( )∫
∞
0
dttf
( ) ( )∫
∞
−
=
0
dtetfsF ts
( ) ( )( ) ( ) ( )∫∫∫ −−−






++





+≤





+−
RRR
z
zd
R
z
ezF
iz
zd
R
z
ezF
iz
zd
R
z
ezFzF
i
zT
T
zTzT
T
γγγ
πππ 2
2
2
2
2
2
1
2
1
1
2
1
1
2
1
Continue with the integral on γR
-
Since FT (z) is entire (analytic in all complex plane we can replace γR
-
with the left
semicircle CL and obtain
( ) ( )
( )
( )
( ) ( )
R
B
R
z
e
z
eBR
z
zd
R
z
ezF
iz
zd
R
z
ezF
i
Tz
Tz
C
zT
T
zT
T
LR
=⋅≤





+=





+
−
∫∫−
2
Re
Re
2
2
2
2
Re2
Re2
1
2
1
1
2
1
π
π
ππ γ

104
SOLO Primes
Donald J. Newman
( 1930 –2007)
and equals F (0).
( )∫
∞
0
dttf
( ) ( )∫
∞
−
=
0
dtetfsF ts
( ) ( )( ) ( ) ( )∫∫∫ −−−






++





+≤





+−
RRR
z
zd
R
z
ezF
iz
zd
R
z
ezF
iz
zd
R
z
ezFzF
i
zT
T
zTzT
T
γγγ
πππ 2
2
2
2
2
2
1
2
1
1
2
1
1
2
1
Continue with the integral on γR
-
Finally we observed that the integral converges to zero uniformly on
compact sets for Re (z) <0 and T→∞, since the integral is the product of
independent of T, and ezT
, which goes to zero uniformly on compact subsets of γR.
( )






+ 2
2
1
R
z
e
z
zF zT
( )






+ 2
2
1
R
z
z
zF
( ) 01
2
1
lim 2
2
=





+∫−
∞→
R
z
zd
R
z
ezF
i
zT
T
γ
π

105
SOLO Primes
Donald J. Newman
( 1930 –2007)
and equals F (0).
( )∫
∞
0
dttf
( ) ( )∫
∞
−
=
0
dtetfsF ts
( ) ( ) ( ) ( )( )
( ) ( )( ) ( ) ( )( )
( )tfB
R
B
z
zd
R
z
ezFzF
iz
zd
R
z
ezFzF
i
z
zd
R
z
ezFzF
i
FF
t
TR
zT
T
zT
T
zT
TT
RR
R
0
2
2
2
2
2
2
max:0
2
1
2
1
1
2
1
1
2
1
00
≥
∞→⇔∞→
=→≤






+−+





+−≤






+−=−
∫∫
∫
−+
γγ
γ
ππ
π
Therefore
( ) ( ) ( )∫
∞
∞→
==
0
00lim tdtfFFT
T
q.e.d.
Return to TOC

106
SOLO
References
Primes
1. Marcus de Sautoy, “The Music of the Primes – Searching to Solve the
Greatest Mystery in Mathematics”, Harper-Collins Publishers, 2003
Internet
http://en.wikipedia.org/wiki/
http://www.mathsisfun.com/prime_numbers.html
http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/giants.pdf
http://plus.maths.org/content/music-primes
N. Levinson, “A Motivated Account of an Elementary Proof of the Prime
Number Theory”, MIT
K. Chandrasekharan, “Lectures on The Riemann Zeta-Function”, Tata
Institute of Fundamental Research, Bombay, 1953
Matt Rosenzweig, “Other Proofs of the Prime Number Theorem”
Jerome Baltzersen, “Hardy’s Theorem and The Prime Number Theorem”,
University of Copenhagen, June 2007
G.B. Arfken, H.J. Weber, “Mathematical Methods for Physicists”, Academic Press,
Fifth Ed., 2001

107
SOLO
References (continue – 1)
Primes
Internet
B.E. Peterson, “Riemann Zeta Funcyion”,
http://people.oregonstate,edu/~peterseb/misc/docs/zeta.pdf
http://mathworld.wolfram.com/RiemannZetaFunctionZeros.html
David Borthwick, “Riemann’s Zeros and the Rhythm of the Primes”, Emory
University, November 18, 2009
Ryan Dingman, “The Riemann Hypothesis”, March 12 2010
Laurenzo Menici, “Zeros of the Riemann Zeta-function on the critical lane”,
Feb. 4 2012, Universita degli Studi, Roma
P.T. Bateman, H.G. Diamond, “A Hundred Years of Prime Numbers”,
http://www.jstor.org
D.J. Newman, “Simple Analytic Proof of the Prime Number Theorem”, The
American Mathematical Monthly, Vol. 87, No. 9 (Nov. 1980), pp. 693- 696
M. Baker, D. Clark, “The Prime Number Theorem”, December 24, 2001
http://www.frm.utn.edu.ar/analisisdsys/material/function_gamma.pdf
K.S. Kedlaya, “Analytic Number Theory”, MIT, Spring 2007, “The Prime
Number Theorem”

108
SOLO
References (continue – 2)
Primes
Internet
D. Miličić, “Notes on Riemann Zeta Function”,
http://www.math.utah.edu/~milicic/zeta.pdf
P. Garrett, “Riemann’ Explicit/Exact formula”, (October 2, 2010),
http://www.math.umn.edu/~garrett/m/mfms/notes_c/mfms_notes_02.pdf
http://homepage.tudelft.nl/11r49/documents/wi4006/gammabeta.pdf
http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/encoding2.htm
Robert B. Ash, “Complex Variables”, Chapter 7: The Prime Number Theorem,
University of Illinois, http://www.math.uluc.edu/~r-ash/CV/CV7.pdf
Physics 116A, “The Riemann Zeta Function”
M. Rosenzweig, “D.J. Newman’s Method of Proof for the Prime Number Theorem”,
M. Rosenzweig, “Other Proofs of the Prime Number Theorem”,
http://people.fas.harvard.edu/~rosenzw/
“Notes on the Riemann Zeta Function”, January 25, 2007

109
SOLO
References (continue –3)
Primes
Internet
A. Granville, K. Soundarajan, “The Distribution of Prime Number”
E.C. Titchmarsh, “The Zeta-Function og Riemann”, Cambridge at the University
Press, 1980
http://www.dartmouth.edu/~chance/chance_news/recent_news/chance_news_10.10.html
http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/encoding2.htm
Hans Riesel, “Prime Numbers and Computer Methods for Factorization”, Chapter
2:“The Primes viewed at Large”,
Prime Numbers and the Riemann Zeta Function « Edwin Chen's Blog
D.R. Heath-Brown, “Prime Number Theory and the Riemann Zeta Function”,
http://eprints.maths.ox.ac.uk/182/1/newton.pdf
http://cage.ugent.be/~jvindas/Talks_files/Introduction_Tauberians_Distributional_A
pproach.pdf

110
Marcus Peter Francis du Sautoy
Prof. Of Mathematics Oxford
University
Return to TOC

March 5, 2015 111
SOLO
Technion
Israeli Institute of Technology
1964 – 1968 BSc EE
1968 – 1971 MSc EE
Israeli Air Force
1970 – 1974
RAFAEL
Israeli Armament Development Authority
1974 – 2013
Stanford University
1983 – 1986 PhD AA

112
SOLO Primes
Definition of O: (E. Landau Definition)
We say that f (x) = O (g (x)) if exists a constant k > 0 such that |f (x)| <
k |g (x)|
Definition of o
We say that f (x) = o (g (x)) when x → a if ( ) ( ) 0/lim =
→
xgxf
ax
Asymptotics
Definition ( ) ( ) axxgxf →,~
means
( ) ( ) ( ) ( ) ( )( ) axxgxgxfisthatxgxf
ax
→+==
→
,,1/lim o
Definitions

113
SOLO Primes
Definition. Let a Function f: Ω → C,
(a)We say that f ϵ C1
(Ω) iff there exists df ϵ C (Ω, M2 (R), a 2x2 matrix-valued
function such that
where d f (s) (h) means that the matrix d f (s) acting on the vector h.
(b) We say that f is Holomorphic on Ω if
exists for all s ϵ Ω and is continuous in Ω. We denote this by f ϵ H (Ω).
A function f ϵ H (C) is called Entire.
( ) ( ) ( )( ) ( ) 0,2
→∈++=+ hRhhohsfdsfhsf
Holomorphic, Entire Functions
( ) ( ) ( )
sw
sfwf
sf
sw −
−
=
→
lim:'
Note that (b) is equivalent to the existence of a function f’ C(Ω) so thatϵ
where f’(s) h is the product between the complex numbers f’(s) and h.
( ) ( ) ( )( ) ( ) 0,2
→∈++=+ hRhhohsfdsfhsf

114
SOLO Primes
Definition.
A Meromorphic Function is a function whose only singularities, except infinity,
are poles.
Meromorphic Functions
E.C. Titchmarch, “Theory of Functions” pg. 284b, 110
A Meromorphic Function in a region if is analytic in the region except at a
finite number of poles. The expression is used in contrast to Holomorphic,
which is some time used instead of Analytic.
Return to TOC

115
SOLO Primes
Mellin Transform
( ){ } ( ) ( )∫
∞
−
==
0
1
xdxfxsFxf s
MM
We can get the Mellin Transform from the two side Laplace Transform
Robert Hjalmar Mellin
( 1854 – 1933)
( ){ } ( ) ( )∫
∞
∞−
−
== xdxfesFxf sx
2LL2
( ){ } ( ) ( )
( ) ( )1
0
11
0
1
+=== ∫∫
∞
−+
∞
−
sFxdxfxxdxfxxxfx ss
MM
( ){ } ( ) ( )∫
∞+
∞−
−
==
ic
ic
s
sdsFx
i
x M
1-
fsfM
π2
1
Example:
{ } ( )sxdexe xsx
Γ== ∫
∞
−−−
0
1
M
( ) x
exf −
=

116
SOLO Primes
Mellin Transform (continue – 1)
( ){ } ( ) ( )∫
∞
−
==
0
1
xdxfxsFxf s
MM
Relation to Two-Sided Laplace Transformation
Robert Hjalmar Mellin
( 1854 – 1933)
tdexdex tt −−
−== ,
Let perform the coordinate transformation
( ) ( )
( ) ( ) ( )∫∫∫
∞
∞−
−−
−∞
∞
−−
∞
−−−−
=−=−= tdeeftdeeftdeefesF tsttstttst
0
1
M
After the change of functions ( ) ( )t
eftg −
=:
( ) ( ) ( ) ( )∫∫
∞
∞−
−
∞
∞−
−−
=== tdetgsGtdeefsF tstst
2LM
Inversion Formula
( ) ( ) ( ) ( ) ( )xfefsdxsF
i
sdesG
i
tg
xe
t
ic
ic
s
exic
ic
ts
tt
=
−
∞+
∞−
−
=∞+
∞−
−−−
==== ∫∫ ML
L
2
1
2
ππ 2
1
2
1

117
SOLO Primes
Properties of Mellin Transform (continue – 2)
( ) ( ){ } ( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) f
kk
k
k
fk
k
k
k
f
z
fk
k
k
f
a
f
f
s
SszsFstf
td
d
t
sksksks
SkszsFkstf
td
d
SzszsFCztft
SssF
sd
d
tft
SsasFaRatf
SsFaataf
SsFtf
HolomorphyofStriptdtftsFtftf
∈+−





−+−−=−
∈−+−−
∈++∈
∈
∈≠∈
>
==>
−−
−
∞
−
∫
M
M
M
M
M
M
M
MM0t,
1
11:
1
,
ln
0,,
0,
11
1
0
1

Original Function Mellin Transform Strip of Convergence

118
SOLO Primes
Properties of Mellin Transform (continue – 3)
( ) ( ){ } ( ) ( )
( ) ( )
( )( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( ) ( ) ( ) 21
0
21
1
0
1
0
1
//
1
1
11:
1
11:
1
ff
t
t
k
f
kk
k
k
k
k
fk
kk
k
k
f
s
SSssFsFxxdxtfxf
sFsxdxf
sFsxdxf
kssss
SssFstf
td
d
t
sksksks
SssFkstft
td
d
SsFtf
HolomorphyofStriptdtftsFtftf



∈⋅
+−
+
−++=
∈−
−+−−=−
∈−−
==>
∫
∫
∫
∫
∞
−
−
∞
∞
−
M2M1
M
M
M
M
M
MM0t,
Original Function Mellin Transform Strip of Convergence
Return to TOC

119
SOLO Primes
( ) ( ) ( ) 1Re
10
1
>=∞<
−
=Γ ∫
∞=
=
−
zxfordt
e
t
zz
t
t
t
z
ς
( ) ∫
∞=
=
−
=Γ
u
u
u
z
du
e
u
z
0
1
Proof:
Gamma Function
Change of variables u=nt ( ) ( )
∫∫
∞=
=
−∞=
=
−
==Γ
t
t
nt
z
z
t
t
nt
z
td
e
t
ntdn
e
nt
z
0
1
0
1
Thus for n=1,2,3,…,N
( )
( )
( ) ∫
∫
∫
∞=
=
−
∞=
=
−
∞=
=
−
=Γ
=Γ
=Γ
t
t
Nt
z
z
t
t
t
z
z
t
t
t
z
z
td
e
t
N
z
td
e
t
z
td
e
t
z
0
1
0
2
1
0
1
1
2
1
1
1

0& >+= xyixz
Summing those equations
for x > 0 ( ) ∫
∞=
=
−






+++=





+++Γ
t
t
z
Ntttzzz
tdt
eeeN
z
0
1
2
1111
2
1
1
1
_________________________________________________


120
SOLO Primes
Proof (continue – 1): 0& >+= xyixz
Since converges only for Re (z)= x > 1, then letting N → ∞, we obtain for x > 1∑
∞
=
−
1n
z
n
Uniform convergence of
( ) ∫
∞=
=
−
∞→






+++=





++Γ
t
t
z
NtttNzz
tdt
eee
z
0
1
2
111
lim
2
1
1
1

 
01
1
1
111
1
2
2
>≥→<=
−
=++
−
δtq
eeee t
q
q
t
q
t
q
t


allows to interchange between limit and the integral:
( ) RatioGoldentd
e
t
td
e
t
td
e
t
z
t
t
t
zt
t
t
zt
t
t
z
zz
=
+
=
−
+
−
=
−
=





++Γ ∫∫∫
∞=
=
−=
+=
−∞=
=
−
2
51
1112
1
1
1
ln2
1ln2
0
1
0
1
φ
φ
φ

∫∫∫
=
+=
−
=
+=
−+==
+=
−






++=
−
=
−
φφφ ln2
0
2
1
ln2
0
1ln2
0
1
11
11
t
t
tt
x
t
t
t
xyixzt
t
t
z
td
ee
ttd
e
t
td
e
t

The first integral gives
The integral diverges for 0 < x ≤ 1, and converges only for x > 1
( ) ( ) ( ) 1Re
10
1
>=∞<
−
=Γ ∫
∞=
=
−
zxfordt
e
t
zz
t
t
t
z
ς

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