The Riemann's Zeta function z(s)

1  Definition

The Zeta function was first introduced by Euler and is defined by

z(s) = ¥
å
n = 1 
 1
ns
 .
 (1)
The series is convergent when s is a complex number with Â(s) > 1. Some special values of z(s) are well known, for example the values z(2) = p2/6, z(4) = p4/90, were obtained by Euler.

In 1859, Riemann had the idea to define z(s) for all complex number s by analytic continuation. This continuation is very important in number theory and plays a central role in the study of the distribution of prime numbers. Several techniques permit to extend the domain of definition of the Zeta function (the continuation is independent of the technique used because of uniqueness of analytic continuation). One can for example start from the Zeta alternating series

za(s) º ¥
å
n = 1 
 (-1)n-1
ns
 ,
defining an analytic function for Â(s) > 0. When the complex number s satisfy Â(s) > 1, we have
za(s) = ¥
å
n = 1 
 1
ns
 - ¥
å
n = 1 
 2
(2n)s
 = z(s) - 2
2s
 z(s).
In other words, we have
z(s) = za(s)
1-21-s
 ,       Â(s) > 1.
 (2)
Since za(s) is defined for Â(s) > 0, this identity (2) permits to define the Zeta function for all complex number s with positive real part, except for s = 1 for which we have a pole.

The extension of the Zeta function to the domain Â(s) £ 0 can also be done (a different technique should be used).

2  The behaviour of z(s) near s = 1

Starting from the formula

1
ns
 = s ó
õ 		¥

n 
 dt
ts+1
 = s ¥
å
k = n 
 ó
õ 		k+1

k 
 dt
ts+1
 ,
a reordering of the summations gives, for Â(s) > 1,
z(s) = s
å
n ³ 1 
å
k ³ n 
 ó
õ 		k+1

k 
 dt
ts+1
 = s
å
k ³ 1 
 æ
ç
è
å
n £ k 
 ó
õ 		k+1

k 
 dt
ts+1
 ö
÷
ø 		= s
å
k ³ 1 
 k ó
õ 		k+1

k 
 dt
ts+1
 .
The last summation writes in the form
z(s) = s ó
õ 		¥

1 
 [t]
ts+1
  dt = s
s-1
 - s ó
õ 		¥

1 
 {t}
ts+1
  dt,
 (3)
where [t] denotes the integer part of t and {t} = t-[t] its fractional part. Notice that the formula (3) is an alternative way to obtain the analytic continuation of z(s) in the half plane Â(s) > 0.

When s = 1, the last integral in (3) is equal to

ó
õ 		¥

1 
 {t}
t2
  dt =
lim
N® ¥ 
 N
å
n = 1 
 ó
õ 		n+1

n 
 t-n
t2
  dt =
lim
N®¥ 
 ó
õ 		N

1 
 dt
t
 - N
å
n = 1 
 1
n+1
 = 1-g,
where g is the Euler constant.

Finally, formula (3) yields the following asymptotic expansion

z(s) = 1
s-1
 + g+ o(1),       (s® 1).
 (4)

This expansion yields interesting results if one computes the expansion obtained by (2) :

z(s) = za(s)
1-21-s
 = za(1)+(s-1)za¢(1)
(s-1)log(2) - (s-1)2log2(2)/2
 + o(1) = za(1)
log(2) (s-1)
 + æ
ç
è 		za¢(1)
log(2)
 + za(1)
2
 ö
÷
ø 		+ o(1).
By comparison with (4), we obtain za(1)/log(2) = 1 and za¢(1)/log(2) + za(1)/2 = g. In other words, we have obtained the classical result
za(1) = ¥
å
n = 1 
 (-1)n-1
n
 = log(2)
and the relation za¢(1) = log(2) (g-za(1)/2) yields the beautiful series
¥
å
n = 1 
 (-1)n log(n)
n
 = log(2) æ
ç
è 		g- log(2)
2
 ö
÷
ø 		.

3  The functional equation

One of the most striking property of the zeta function, discovered by Riemann himself, is the functional equation :

z(s) = c(s)z(1-s),       c(s) = 2s ps-1 sin æ
ç
è 		ps
2
 ö
÷
ø 		G(1-s).
 (5)
The G(s) function is the Euler function.

From the continuation of z(s) in the half plane Â(s) > 0, notice that the functional equation is gives the analytic continuation of z(s) to the whole complex plane.

This analytic continuation can be obtained in several ways (see [2] for a list of seven methods to prove the functional equation).

The complement formula of the Gamma function (see The Gamma function G(x)) entails the formula

c(s)c(1-s) = 1,
which gives a symetry of the functional equation with respect to the line Â(s) = 1/2.

4  Relation with series of primes

Let pn denote the n-th prime (p1 = 2, p2 = 3, p3 = 5, ¼). We have

N
Õ
i = 1 
 æ
ç
è 		1+ 1
pis
 + 1
pi2s
 +¼ ö
÷
ø 		= 1+ 1
n1s
 + 1
n2s
 +¼
where n1, n2, ¼ are those integers none of whose prime factors exceed P = pN. Since all integers up to P are of this form, it follows that
ê
ê
ê 		z(s) - N
Õ
i = 1 
 æ
ç
è 		1- 1
pis
 ö
÷
ø 		-1

 
 ê
ê
ê 		= ê
ê
ê 		z(s) - 1- 1
n1s
 - 1
n2s
 -¼ ê
ê
ê 		£ 1
(P+1)Â(s)
 + 1
(P+2)Â(s)
 + ¼
Letting N® ¥, we finally obtain the beautiful Euler's product
z(s) =
Õ
p prime 
 1
1-p-s
 .

Euler's product makes the Riemann zeta function interesting in the theory of prime numbers. Combining this identity with properties of z(s) gives interesting information about the series of primes. The most famous result of this kind is due to Hadamard and De La Vallée Poussin, who independently proved in 1896 that

p(x) ~ x
log(x)
 ,
where p(x) denote the number of primes not exceeding x. This result is known as the prime number theorem.

5  The Riemann hypothesis

It was conjectured by Riemann that all the non trivial complex zeros s of z(s) lie on the critical line Â(s) = 1/2. This conjecture is known as the Riemann hypothesis has never been proved or disproved. The importance of the Riemann hypothesis lies in the estimation of p(x), the number of primes not exceeding x : If Riemann hypothesis is true, then we have

p(x) = ó
õ 		x

2 
 dt
log(t)
 + O(x1/2+e),      (for all e > 0).
In other words, the p(x) = ò2x dt/log(t) estimation is good with an error of the order x1/2. The Riemann hypothesis is related with this error term, and the best bounds known today are far from this result.

The Riemann hypothesis is one of the most famous unsolved mathematical problems. Numerical computations have been made to check the Riemann hypothesis on the first 1.5 109 zeros [3]. Other partial computations at larger indexes of zeros have also been made (A. M. Odlyzko).

References

[1]
H. M. Edwards, Riemann's Zeta Function, Academic Press, 1974.

[2]
E. C. Titchmarsh, The theory of the Riemann Zeta-function, Oxford Science publications, second edition, revised by D. R. Heath-Brown (1986).

[3]
J. van de Lune, H. J. J. te Riele and D. T. Winter, On the zeros of the Riemann zeta function in the critical strip, IV. Math. Comp. 46 (1986), 667-681.

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