Riemann zeta function
1 Definition
The Riemann zeta function is defined to be the complex valued
function
given by the series
ζ(s):= | (1) |
which is valid (in fact, absolutely convergent) for all complex
numbers with . We list here some of the key
properties [1] of the zeta function
.
-
1.
For all with , the zeta function satisfies the Euler product formula
(2) where the product
is taken over all positive integer primes , and converges uniformly in a neighborhood of .
-
2.
The zeta function has a meromorphic continuation to the entire complex plane with a simple pole
at , of residue
, and no other singularities.
- 3.
2 Distribution of primes
The Euler product formula (2) given above expresses the
zeta function as a product over the primes , and
consequently provides a link between the analytic properties of the
zeta function and the distribution of primes in the integers. As the
simplest possible illustration of this link, we show how the
properties of the zeta function given above can be used to prove that
there are infinitely many primes.
If the set of primes in were finite, then the Euler product formula
would be a finite product, and consequently would exist and would equal
But the existence of this limit contradicts the fact that has a pole at , so the set of primes cannot be finite.
A more sophisticated analysis of the zeta function along these lines
can be used to prove both the analytic prime number theorem
and
Dirichlet’s theorem
on primes in arithmetic progressions11In the case of arithmetic progressions
, one also needs to examine the closely related Dirichlet –functions in addition to the zeta function itself.. Proofs of
the prime number theorem can be found in [2]
and [5], and for proofs of Dirichlet’s theorem on primes
in arithmetic progressions the reader may look in [3]
and [7].
3 Zeros of the zeta function
A nontrivial zero of the Riemann zeta function is defined to be a root of the zeta function with the property that . Any other zero is called trivial zero of the zeta function.
The reason behind the terminology is as follows. For complex numbers
with real part greater than 1, the series definition (1)
immediately shows that no zeros of the zeta function exist in this
region. It is then an easy matter to use the functional
equation (3) to find all zeros of the zeta function
with real part less than 0 (it turns out they are exactly the values
, for a positive integer). However, for values of with
real part between 0 and 1, the situation is quite different, since we
have neither a series definition nor a functional equation to fall
back upon; and indeed to this day very little is known about the
behavior of the zeta function inside this critical strip
of the
complex plane.
It is known that the prime number theorem is equivalent to the
assertion that the zeta function has no zeros with or
. The celebrated Riemann hypothesis asserts that all nontrivial zeros of the zeta function satisfy the much more precise equation . If true, the hypothesis
would have profound
consequences on the distribution of primes in the
integers [5].
References
- 1 Lars Ahlfors, Complex Analysis, Third Edition, McGraw–Hill, Inc., 1979.
- 2 Joseph Bak & Donald Newman, Complex Analysis, Second Edition, Springer–Verlag, 1991.
-
3
Gerald Janusz, Algebraic Number Fields
, Second Edition, American Mathematical Society, 1996.
-
4
Serge Lang, Algebraic Number Theory
, Second Edition, Springer–Verlag, 1994.
- 5 Stephen Patterson, Introduction to the Theory of the Riemann Zeta Function, Cambridge University Press, 1988.
- 6 B. Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/
-
7
Jean–Pierre Serre, A Course in Arithmetic
, Springer–Verlag, 1973.
Title | Riemann zeta function |
Canonical name | RiemannZetaFunction |
Date of creation | 2013-03-22 12:38:01 |
Last modified on | 2013-03-22 12:38:01 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 18 |
Author | alozano (2414) |
Entry type | Definition |
Classification | msc 11M06 |
Synonym | function |
Related topic | AnalyticContinuationOfRiemannZeta |
Related topic | DedekindZetaFunction |
Related topic | DirichletSeries |
Related topic | EulerProduct |
Related topic | Complex |
Related topic | EulerProductFormula2 |
Related topic | HarmonicSeriesOfPrimes |
Defines | Euler product formula |
Defines | Riemann hypothesis |