Riemann zeta function

In mathematics, the Riemann zeta function, named after Bernhard Riemann, is one of the most important special functions. Its generalizations have important applications to number theory, arithmetic geometry, graph theory, and dynamical systems, to name a few examples. The Riemann zeta function in particular gained prominence when it was shown to have a connection with the distribution of the prime numbers. The most important result related to the Riemann zeta function is the Riemann hypothesis, which was the 8th of Hilbert's Problems, and is one of the seven Millenium Prize Problems presented by the Clay Institute of Mathematics. As such, anyone who determines its truth or falsity is entitled to $1 million (U.S.)

Definition

The Riemann zeta function is a meromorphic function defined for complex numbers with real part $\scriptstyle \Re (s)>1$ by the infinite series

$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}$

and then extended to all other complex values of s except s = 1 by analytic continuation. The function is holomorophic everywhere except for a simple pole at s = 1.

Euler's product formula for the zeta function is

$\zeta (s)=\prod _{p\ \mathrm {prime} }{\frac {1}{1-p^{-s}}}$

(the index p running through the set of prime numbers).

The celebrated Riemann hypothesis is the conjecture that all non-real values of s for which ζ(s) = 0 have real part 1/2. The problem of proving the Riemann hypothesis is the most well-known unsolved problem in mathematics.

History

The origin of the Riemann zeta function can be traced to the Basel Problem. The solution to this problem states that

$\zeta (2)=\sum _{1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}.$

In deriving this identity, Leonard Euler also found the sums of the series $\zeta (2k)$ for $\scriptstyle k=1,\ldots ,13$ These computations contain implicitly the germ of the idea of the zeta function. According to Andre Weil, these and related results due to Euler remained as "mere curiosities, and virtually unknown, until they received new life at the hands of Riemann in 1859".

In an eight page paper, Riemann catapulted both himself and his namesake to worldwide renown. The paper includes the general definition of the zeta function (including the first use of the symbol $\zeta$ to denote it) and the proof of its analytic continuation, the functional equation (see below), as well as results relating the function to the distribution of prime numbers and of course, the Riemann hypothesis. Since then, the function and its relatives have found applications in myriad research papers in a vast array of fields.