Definition:Riemann Zeta Function

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The analytic continuation of $\zeta$

Definition

The Riemann $\zeta$ (zeta) function is the complex function defined on the half-plane $\map \Re s > 1$ as the series:

$\ds \map \zeta s = \sum_{n \mathop = 1}^\infty \frac 1 {n^s}$


Critical Strip

Let $s = \sigma + i t$.

The region defined by the equation $0 < \sigma < 1$ is known as the critical strip.


Critical Line

Let $s = \sigma + i t$.

The line defined by the equation $\sigma = \dfrac 1 2$ is known as the critical line.

Hence the popular form of the statement of the Riemann Hypothesis:

All the nontrivial zeroes of the Riemann zeta function lie on the critical line.


Analytic Continuation

By Analytic Continuations of Riemann Zeta Function, $\zeta$ has a unique analytic continuation to $\C \setminus \set 1$.


This analytic continuation is still called the Riemann $\zeta$ function and still denoted $\zeta$.


Zeroes

The zeroes of the Riemann $\zeta$ function are, according to the conventional definition of the zero of a (complex) function, the points $s \in \C$ such that:

$\map \zeta s = 0$


Trivial Zeroes

The trivial zeroes of the Riemann $\zeta$ function are the strictly negative even integers :

$\set {n \in \Z: n = -2 \times k: k \in \N_{\ne 0} } = \set {-2, -4, -6, \ldots}$


Nontrivial Zeroes

The nontrivial zeroes of the Riemann $\zeta$ function are the zeroes of $\zeta$ that are not trivial.


Also known as

The Riemann zeta function is also often known as just the zeta function.

Some sources use hyphens: Riemann zeta-function or zeta-function

Some sources style it as Riemann's zeta function.


Also see

  • Results about the Riemann $\zeta$ function can be found here.


Special values


Generalizations


Source of Name

This entry was named for Georg Friedrich Bernhard Riemann.


Historical Note

Investigation of what is now known as the Riemann zeta function goes back as far as Leonhard Paul Euler, who showed in $1744$ that $\ds \sum_{n \mathop = 1}^\infty \frac 1 {n^s} = \prod_{\text {$p$ prime} } \frac 1 {1 - p^{-s} }$.


The Riemann zeta function was discussed by Bernhard Riemann in his $1859$ article Ueber die Anzahl der Primzahlen under einer gegebenen Grösse.

In that paper he made several statements about that function.

Studying it led Jacques Salomon Hadamard to his proof of the Prime Number Theorem.


All of the statements made in that paper have now been proved except for one.

The last remaining statement which has not been resolved is the Riemann Hypothesis.


When Riemann first investigated this function, he used $s$ instead of the more typical complex variable $z$.

To this day, $s = \sigma + i t$ is still typically used as the complex variable in investigations of the zeta function.


Sources

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