The **Riemann zeta function** is defined for any complex number *s* with real part > 1 as:

*s*: Re(

*s*) > 1}, this infinite series converges and defines a holomorphic function. (In that expression, Re means the real part of a number.) Bernhard Riemann realized that the zeta function can be extended by analytic continuation in a unique way to a holomorphic function defined for

**all**complex numbers

*s*with

*s*≠ 1. It is this function that is the object of the Riemann hypothesis.

The connection between this function and prime numbers was already realized by Leonhard Euler:

*p*. This is a consequence of the formula for the geometric series and the Fundamental Theorem of Arithmetic.

The zeros of ζ(*s*) are important because certain path integrals involving the function ln(1/ζ(*s*)) can be used to approximate the prime counting function π(*x*) (see prime number theorem). These path integrals are computed with the residue theorem and hence knowledge of the integrand's singularities is required.

The zeta function satisfies the following functional equation:

*s*in

**C**- {0,1}. Here, Γ denotes the Gamma function. This formula is used to construct the analytic continuation in the first place. At

*s*= 1, the zeta function has a simple pole with residue 1.

Euler was also able to calculate ζ(*2k*) for even integers *2k* using the formula

*B*

_{2k}are the Bernoulli numbers. From this one sees that ζ(2) = π

^{2}/6, ζ(4) = π

^{4}/90, ζ(6) = π

^{6}/945 etc. These give well-known infinite series for &pi. For odd integers the case is not so simple. Ramanujan did some great work about this.

One can express the reciprocal of the zeta function using the Möbius function μ(*n*) as follows:

*s*with real part > 1. This, together with the above expression for ζ(2), can be used to prove that the probability of two random integers being coprime is 6/π

^{2}.

Although Riemann's zeta function is thought of by mathematicians as being relevant to the "purest" of mathematical disciplines, number theory, it also occurs in applied statistics; see Zipf's law and Zipf-Mandelbrot law.