Riemann zeta function

In mathematics, the Riemann zeta function is a function which is of paramount importance in number theory, because of its relation to the distribution of prime numbers. It also has applications in physics.

Definition

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

In the region {s in C: Re(s) > 1}, this infinite series converges and defines a holomorphic function. Bernhard Riemann realized that the zeta function can be extended by analytic continuation in a unique way to a holomorphic function ζ(s) defined for all complex numbers s with s ≠ 1. It is this function that is the object of the Riemann hypothesis.

Relationship to prime numbers

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

an infinite product extending over all prime numbers 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). These path integrals are computed with the residue theorem and hence knowledge of the integrand's singularities is required.

Basic properties

The zeta function satisfies the following functional equation:

valid for all 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

where B_2k are the Bernoulli numbers. From this one sees that ζ(2) = π²/6, ζ(4) = π⁴/90, ζ(6) = π⁶/945 etc. (sequence A046988 /A002432 in OEIS). These give well-known infinite series for π. 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:

for every complex number 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/π².

Applications

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