In mathematics, a Dirichlet series, one of a number of concepts named in honor of Johann Peter Gustav Lejeune Dirichlet, is a series of the form Mathematics is the study of quantity, structure, space and change. ... Peter Gustav Lejeune Dirichlet. ... In mathematics, a series is the sum of a sequence of terms. ...

The most famous of Dirichlet series is

which is the 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. ...

Other Dirichlet series are:

where μ(n) is the Möbius function, The classical Möbius function is an important multiplicative function in number theory and combinatorics. ...

where φ(n) is the totient function, and In number theory, the totient φ(n) of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n. ...

where σ_a(n) is the divisor function In mathematics the divisor function σa(n) is defined as the sum of the ath powers of the divisors of n, or The notations d(n) and (the tau function) are also used to denote σ0(n), or the number of divisors of n. ...

Analytic properties of Dirichlet series

Given a sequence {a_n}_{n ∈ N} of complex numbers we try to consider the value of

as a function of the complex variable s. In order for this to make sense, we need to consider the convergence properties of the above infinite series: In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...

Theorem. Suppose {a_n}_{n ∈ N} is a bounded sequence of complex numbers. Then the above infinite series for f converges absolutely on the open half-plane of s such that Re(s) > 1. In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. ... In mathematics, a series is a sum of a sequence of terms. ...

If the set of sums a_n + a_{n + 1} + ... + a_{n + k} is bounded for n and k ≥ 0, then the above infinite series converges on the open half-plane of s such that such that Re(s) > 0.

In both cases f is an analytic function on the corresponding open half plane. In mathematics, an analytic function is one that is locally given by a convergent power series. ...

In general the abscissa of convergence of a Dirichlet series is the intercept on the real axis of the vertical line in the complex line, such that there is convergence to the right of it, and divergence to the left. This is the analogue for Dirichlet series of the radius of convergence for power series. The Dirichlet series case is more complicated, though: absolute convergence and uniform convergence may occur in distinct half-planes. In mathematics, the radius of convergence of a power series where the center a and the coefficients cn are complex numbers (which may, in particular, be real numbers) is the nonnegative quantity r (which may be a real number or ∞) such that the series converges if and diverges if In... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... In mathematics, a series is a sum of a sequence of terms. ... In mathematical analysis, a sequence { fn } of functions converges uniformly to a limiting function f if the speed of convergence of fn(x) to f(x) does not depend on x. ...

In many cases, the analytic function associated with a Dirichlet series has an analytic extension to a larger domain. This is the case for the zeta function:

Theorem. The zeta function has a meromorphic extension to C with a unique pole at s = 1.

One of the most important open conjectures of mathematics — the Riemann hypothesis — concerns the zeroes of the zeta function. Mathmatical and Non-Mathamatical Definitions In mathematics, a conjecture is a mathematical statement which has been proposed as a true statement, but which no one has yet been able to prove or disprove. ... In mathematics, the Riemann hypothesis (aka Riemann zeta hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous of all unsolved problems. ...

See also

• Dirichlet

Johann Peter Gustav Lejeune Dirichlet (February 13, 1805 - May 5, 1859) was a German mathematician credited with the modern formal definition of a function. ...

External links

Dirichlet series on PlanetMath. PlanetMath is a free, collaborative, online mathematics encyclopedia. ...