Riemann zeta function

Riemann zeta function
Name	Riemann zeta function
Caption	Absolute value of the Riemann zeta function in the complex plane.
General definition	$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$ for $\Re(s) > 1$
Domain	$\mathbb{C} \setminus \{1\}$
Codomain	$\mathbb{C}$
Zero	-2, -4, -6, ... (trivial); $\Re(s) = \tfrac{1}{2}$ (conjectured, non-trivial)
Pole	$s = 1$ (simple pole)
Discoverer	Bernhard Riemann
Year	1859
Related functions	Dirichlet L-function, Dedekind zeta function, Hurwitz zeta function

Contents

Definition
Euler product formula
Special values
Functional equation
Zeros and the Riemann hypothesis
Applications

Riemann zeta function is a central object of study in analytic number theory and complex analysis, originally introduced by Leonhard Euler and later profoundly developed by Bernhard Riemann. Its properties are deeply connected to the distribution of prime numbers, most famously through the unsolved Riemann hypothesis. The function extends the harmonic series to the complex plane and exhibits a profound symmetry described by its functional equation.

Definition

The Riemann zeta function is most commonly defined for complex numbers with a real part greater than one by the Dirichlet series $\zeta(s) = \sum_{n=1}^\infty n^{-s}$ . This series converges absolutely in the half-plane $\Re(s) > 1$ . To define it for other values of $s$ , the function is analytically continued to the entire complex plane except for a simple pole at $s = 1$ . This continuation is a meromorphic function with residue 1 at its pole. The process of analytic continuation is often achieved via the functional equation or integral representations involving the Gamma function.

Euler product formula

For $\Re(s) > 1$ , the Riemann zeta function possesses a fundamental representation as an infinite product over all prime numbers, known as the Euler product: $\zeta(s) = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}$ . This identity, discovered by Leonhard Euler, provides a direct link between the function and the primes, as it equates a sum over all natural numbers to a product over primes. It is a cornerstone in proving the infinitude of primes and is a prototype for similar product formulas associated with Dirichlet L-functions and Dedekind zeta functions.

Special values

The Riemann zeta function takes on notable values at both positive and negative integers, as well as at even positive integers. At negative integers, $\zeta(-n) = -\frac{B_{n+1}}{n+1}$ for $n \ge 0$ , where $B_k$ are Bernoulli numbers; these are the so-called trivial zeros. At positive even integers, $\zeta(2n) = (-1)^{n+1} \frac{B_{2n} (2\pi)^{2n}}{2(2n)!}$ , with famous results like $\zeta(2) = \frac{\pi^2}{6}$ , a solution to the Basel problem. The value at $s=1$ corresponds to the divergent harmonic series, manifesting as the pole.

Functional equation

A profound symmetry of the Riemann zeta function is encapsulated in its functional equation, which relates its values at $s$ and $1-s$ . A common symmetric form is $\zeta(s) = 2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1-s) \zeta(1-s)$ , where $\Gamma$ is the Gamma function. This equation, proven in Riemann's seminal 1859 paper "On the Number of Primes Less Than a Given Magnitude", is valid for all complex $s$ and reveals that the function's behavior is determined by its values in a critical strip. It implies the existence of the trivial zeros at negative even integers.

Zeros and the Riemann hypothesis

The zeros of the Riemann zeta function are of paramount importance. The trivial zeros occur at all negative even integers. The non-trivial zeros lie in the critical strip $0 < \Re(s) < 1$ and are symmetrically distributed about the real axis and the critical line $\Re(s) = \frac{1}{2}$ . The unproven Riemann hypothesis, one of the Clay Millennium Prize Problems, conjectures that all non-trivial zeros have real part exactly one-half. The distribution of these zeros is intimately connected to the error term in the prime number theorem, as studied in the work of John Edensor Littlewood and Atle Selberg.

Applications

The Riemann zeta function has vast applications, primarily in analytic number theory. It is instrumental in proving the prime number theorem, which describes the asymptotic distribution of primes. Its generalizations, like Dirichlet L-functions, are used in proofs of Dirichlet's theorem on arithmetic progressions. The function also appears in quantum field theory and statistical mechanics, particularly in calculations involving Bose–Einstein statistics and Casimir effect. Furthermore, its special values have deep connections to K-theory and p-adic analysis.

Category:Zeta and L-functions Category:Analytic number theory Category:Complex analysis Category:Millennium Prize Problems