The Riemann Zeta-function z(s) : generalities
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The Zeta function was first introduced by Euler and is defined by
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 independant of the technique used
because of uniqueness of analytic continuation).
One can for example start from the
Zeta alternating series (also called the Dirichlet eta function)
h(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
h(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) = |
h(s)
1-21-s
|
, Â(s) > 1. |
| (2) |
Since h(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 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) = |
h(s)
1-21-s
|
= |
h(1)+(s-1)h¢(1)
(s-1)log(2) - (s-1)2log2(2)/2
|
+ o(1) |
| |
| = |
h(1)
log(2) (s-1)
|
+ |
æ è
|
h¢(1)
log(2)
|
+ |
h(1)
2
|
ö ø
|
+ o(1). |
| |
|
By comparison with (4), we obtain
h(1)/log(2)=1 and h¢(1)/log(2) + h(1)/2 = g.
In other words, we have obtained the classical result
h(1) = |
¥ å
n=1
|
|
(-1)n-1
n
|
= log(2) |
|
and the relation h¢(1) = log(2) (g-h(1)/2)
yields the beautiful series
|
¥ å
n=1
|
(-1)n |
log(n)
n
|
= log(2) |
æ è
|
g- |
log(2)
2
|
ö ø
|
. |
|
Generalized Euler constants
The expansion (4) can be continued by writing
z(s) = |
1
s-1
|
+ |
¥ å
n=0
|
|
(-1)n
n!
|
gn(s-1)n. |
|
The constants gn can be proved to satisfy
gn = |
lim
m ® ¥
|
|
m å
k=1
|
|
logn k
k
|
- |
logn+1 m
n+1
|
. |
|
These formula generalize the Euler constant definition (corresponding
to the case n=0) and for that reason, the constants gn are
often called the generalized Euler constants. They were also called
Stieljes constants as they were studied by Stieljes.
General informations about these constants can be found on
Eric
Weissten's world of mathematics site.
3 More on the analytic continuation of zeta
|
We have seen two different ways to extend the analytic continuation of
z(s) to the domain Â(s) > 0 (see (2)
and (3)).
In fact, the Riemann Zeta function can be analytically continued to
the whole complex plane. A possible way (see [2] for
example) to obtain this fundamental property
is to generalize the approach of (3) by using
Euler-Maclaurin sum formula applied to the function f(x) = x-s with
order q, which yields
|
z(s) = |
1
s-1
|
+ |
1
2
|
+ |
q å
r=2
|
|
Br
r!
|
s(s+1)¼(s+r-2) |
| |
| |
|
- |
1
q!
|
s(s+1)¼(s+q-1) |
ó õ
|
+¥
1
|
Bq({x}) x-s-q dx, |
| | (5) |
|
where the Br are the Bernoulli numbers and Bq({x}) the Bernoulli
polynomials evaluated at the fractionnal value {x}=x-[x].
The integral does not only converge for Â(s) > 1, but also for
Â(s) > 1-q. Thus formula (5) provides the
analytic continuation of z(s) to the half plane
Â(s) > 1-q. Since q can be choosen arbitrarily large, we conclude that
z(s) is analytic in the whole complex plane with a simple pole
at s=1.
Formula (5) also permits to easily obtain values
of z(s) at some special points. For example, when s=0, only
the two first terms of the formula do not vanish, which yields
When s=-k with k a positive integer, choosing q=k+1 in
formula (5) leads to
| |
|
- |
1
k+1
|
+ |
1
2
|
- |
q å
r=2
|
|
Br
r!
|
k(k-1)¼(k-r+2) |
| |
| |
|
- |
1
k+1
|
|
q å
r=0
|
|
æ è
|
k+1
r
|
ö ø
|
Br = - |
Bk+1(1)
k+1
|
= - |
Bk+1
k+1
|
. |
| |
|
We have used the fact that the Bernoulli numbers Bk vanish when k is odd
except for k=1 for which B1=-1/2, and the representation of Bernoulli
polynomials in terms of Bernoulli numbers.
We conclude that z(s) vanishes for even negative integers (the
corresponding zeros are called the "trivial zeros" of z(s))
and that for positive integers m,
4 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 è
|
ps
2
|
ö ø
|
G(1-s). |
| (7) |
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 another way to obtain the analytic
continuation of z(s) to the whole complex plane.
The complement formula of the Gamma function
(see The Gamma function G(x)) entails
the formula
which gives a symetry of the functional equation with respect to the line
Â(s)=1/2.
A proof of the functional equation
The functional equation (7) is fascinating and
mathematicians afforded numerous different proofs of it. Riemann
himself gave several methods ; later, mathematicians like Hardy,
Siegel and others, enriched the list of proofs (see for
example [4], where seven different methods are presented).
We present here briefly one proof of the functional equation, extracted
from [2] where we will see that the functional equation
is strongly related to Fourier series.
The starting point is formula (5) applied with q=2
z(s) = |
1
s-1
|
+ |
1
2
|
+ |
B2
2
|
s - |
s(s+1)(s+2)
6
|
|
ó õ
|
+¥
1
|
B3({x}) x-s-3 dx. |
| (8) |
The formula is valid for Â(s) > -2.
The Bernoulli polynomial B3(x)=x(x-1/2)(x-1) has a factor x, thus the
integral ò01 B3(x) x-s-3 is convergent for Â(s) < -1.
The value of this integral is easily obtained thanks to repeted integration by
part, and plugging it into (8) leads to the simple relation
z(s) = - |
s(s+1)(s+2)
6
|
|
ó õ
|
+¥
0
|
B3({x}) x-s-3 dx (-2 < Â(s) < -1). |
| (9) |
We now use the Fourier series
B3({x}) = 12 |
¥ å
n=1
|
|
sin(2npx)
(2np)3
|
|
|
and plug it into (9), which gives
z(s) = - 2 s(s+1)(s+2) |
¥ å
n=1
|
|
1
(2np)3
|
|
ó õ
|
¥
0
|
|
sin(2npx)
xs+3
|
dx, (-2 < Â(s) < -1). |
|
We omit the justification of term-by-term integration, which can be
found in [2]. The latest formula rewrites as
z(s) = - 2 s(s+1)(s+2) |
¥ å
n=1
|
|
1
(2np)1-s
|
|
ó õ
|
¥
0
|
|
sin(y)
ys+3
|
dy, (-2 < Â(s) < -1). |
|
The integral ò0¥ sin(y) y-s-3 dy is classic and its value is
G(-s-2) sin(p(-s-2)/2) (see [2] for example), thus
| |
|
2 s(s+1)(s+2) G(-s-2) sin |
p(s+2)
2
|
|
1
(2p)1-s
|
|
¥ å
n=1
|
|
1
n1-s
|
, |
| |
| |
|
2s ps-1 G(1-s) sin |
ps
2
|
z(1-s) (-2 < Â(s) < -1). |
| |
|
This is the functional equation, proved for complex values of s in the
strip -2 < Â(s) < -1, and it is valid in
the whole complex plane by analytic continuation.
5 Values of z(s) at positive integers
|
The functional equation (7) together with
identity (6) yields the other identity, valid for
positive integers m
z(2m) = |
4m (-1)m-1 B2m p2m
2 (2m)!
|
|
| (10) |
Notice that this famous formula can be obtained independantly of the functional
equation (from Fourier series of Bernoulli polynomials for example) and permits
with (6) to check that the functional equation is
fulfilled for even positive integers values of s.
This formula applied with the first values of m give the famous
special values
z(2) = |
p2
6
|
, z(4) = |
p4
90
|
, z(6) = |
p6
945
|
, z(8) = |
p8
9450
|
. |
|
Values of z(s) at odd positive integers
No equivalent formula of z(m) are known for odd positive values of
m, and it is strongly expected that they do not exist. Numerical experiments
show for example that if z(3) have the form p3 p/q, then the
denominator q has a very large number of digits.
The first historical result obtained on values of z(s) at odd positive
integers is due to Apery who proved in 1978 that z(3) is irrational
(for that reason, z(3) is now sometimes called the Apery constant).
It is not known if z(3) is transcendantal.
Apery's proof for z(3) does not generalize for z(5),
z(7), ¼, and it is not known if any of these
constants are irrational or not. However, in 2000, Rivoal
(see [3]) stated that
an infinity of the values z(2m+1) for positive integer m are
irrational (without precising which of those values are irrational).
Extending the method of Rivoal, other more precise results have been
found. The most striking is due to Zudilin
(see [5]) who proved that one of the numbers z(5),
z(7), z(9), z(11) is irrational. Other results like
"for every odd integer b, one of the numbers z(b+2),
z(b+4), ¼, z(8b-3), z(8b-1) is irrational"
can be found in [6] for example.
6 Relation with series of primes
|
Let pn denote the n-th prime number (p1=2, p2=3, p3=5,
¼).
We have
|
N Õ
i=1
|
|
æ 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
|
z(s) - |
N Õ
i=1
|
|
æ è
|
1- |
1
pis
|
ö ø
|
-1
|
ê ê
|
= |
ê ê
|
z(s) - 1- |
1
n1s
|
- |
1
n2s
|
-¼ |
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 independantly proved in 1896 that
where p(x) denote the number of primes not exceeding x.
This result is known as the prime number theorem.
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. It is undoubtely the most celebrated problem in
mathematics, not only because it has gone unsolved for so long (more
than one century) but also because it has some important consequences
in the distribution of primes.
A lot of informations about Riemann Hypothesis can be found in the
following
link.
7.1 Consequences of the Riemann hypothesis
The most important consequence 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(x1/2logx), |
| (11) |
whereas the error bound on the approximation of p(x) by
ò2x dt/log(t) obtained without the Riemann hypothesis is
p(x) = |
ó õ
|
x
2
|
|
dt
log(t)
|
+ O |
æ è
|
x exp |
æ è
|
- |
A (logx)3/5
(loglogx)1/5
|
ö ø
|
ö ø
|
. |
| (12) |
Estimate (11) is much better
than (12) because it states that the order of
approximation of p(x) by ò2x dt/log(t) is better than
O(x1/2+e) whereas the second one is worst than
O(x1-e) for all e > 0.
In fact, the quality of the approximation is
directly related to the "size" of the
free zero region near the line Â(s)=1. The
estimate (12) have been obtained in 1958 when
Vinogradov and Korobov enlarged the known free zero region of
z(s) near Â(s)=1.
Among the most famous other consequences of the Riemann hypothesis (RH), we
have :
- The Lindelöf hypothesis : if the RH is true, then when the
positive real number t goes to infinity, we have
z(1/2+it) = O(te) for all e > 0 when
t®¥. The best estimate known
today without the Riemann hypothesis is
z(1/2+it)=O(tA+e) with A=139/858=0.162004¼
(Kolesnik).
- The Möbius function mean value estimate : if the RH is true,
then M(x) = O(x1/2+e) for all e > 0, where
M(x)=ån £ x m(n) with m(n) the Möbius
function, defined by m(n)=(-1)k if n is the product of k
distinct primes, and m(n)=0 if n can be divided by the square
of some prime.
- Bounds on the values of Zeta on the vertical
line s = 1 . Under the RH one has
z(1+it) = O(loglogt), |
1
z(1+it)
|
= O(loglogt). |
|
Without the RH, the best bound known today are
z(1+it) = O(logt/loglogt) and 1/z(1+it) = O(logt/loglogt).
7.2 Attempts to prove the RH
Numerous mathematicians tried to prove or disprove the RH. Thousands of
(wrong) proofs have been afforded for more than one century (see the
following link
proposed
proofs of the Riemann Hypothesis for more details).
for a
On the other hand, some results are known about the zeros of the
Zeta-function, the most classical are discussed
in Distribution of the zeros of the Riemann Zeta
function.
Another approach consists in numerical computations, which could
disprove the Riemann hypothesis by exhibiting a zero off the critical
line. For example, it has been numerically checked that the first
1012 zeros are on the critical line. In addition to the pure RH
verification, Odlyzko in [1] computed statitics on the
distribution of zeros, that could orientate possible proofs.
More about numerical computations on the zeros of zeta is discussed
in Numerical computations about the zeros of
the zeta function.
In 2000, Clay Mathematics Institute offered a one million dollars
prize for proof of the Riemann hypothesis. Interestingly, disproof of
the Riemann hypothesis (e.g., by using a computer to actually find a
zero off the critical line), does not earn the one million dollars award.
References
- [1]
-
A. M. Odlyzko.
The 1022-th zero of the riemann zeta function.
In M. van Frankenhuysen and M. L. Lapidus, editors, Dynamical,
Spectral, and Arithmetic Zeta Functions, number 290 in Contemporary Math.
series, pages 139-144. Amer. Math. Soc., 2001.
- [2]
-
H. Rademacher.
Topics in Analytic number theory.
Springer Verlag, 1973.
- [3]
-
T. Rivoal.
La fonction zêta de riemann prend une infinité de valeurs
irrationnelles aux entiers impairs.
C. R. Acad. Sci. Paris Sér. I Math., 2000.
- [4]
-
E. C. Titchmarsh.
The theory of the Riemann Zeta-function.
Oxford Science publications, second edition, 1986.
revised by D. R. Heath-Brown.
- [5]
-
W. Zudilin.
One of the numbers z(5), z(7), z(9), z(11) is
irrational.
Uspekhi Mat. Nauk, (56:4):149-150, 2001.
English transl. Russian Math. Surveys 56 (2001) 774-776.
- [6]
-
W. Zudilin.
Irrationality of values of riemann's zeta function.
Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv.
Math.], 2002.
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