Dirichlet series

In mathematics, a Dirichlet series is an infinite series whose terms involve successive positive integers raised to powers of a variable, typically with integer, real or complex coefficients. If the series converges, its value determines a function of the variable involved.

Formally, let s be a variable and $a_{n}$ be a sequence of real or complex coefficients. The associated Dirichlet series is

$\sum _{n=1}^{\infty }a_{n}n^{-s}.\,$.

Over the complex numbers the series will have an abscissa of convergence S, a real number with the property that the series converges for all complex numbers s with real part $\Re s>S$ and that S is the "smallest" number with this property (infimum of all numbers with this property). If the series converges for all complex numbers, we formally say that the abscissa of convergence is infinite.

For example

$\sum n^{-s}$ converges for all $\Re s>1$, but diverges for $s=1$ and so has abscissa of convergence 1.

In the half-plane to the right of the abscissa of convergence, a Dirichlet series determines an analytic function of s.

Dirichlet series may be added and multiplied. If $\sum a_{n}n^{-s}$ and $\sum b_{n}n^{-s}$ are Dirichlet series, we may define their sum and product

$\left(\sum a_{n}n^{-s}\right)+\left(\sum b_{n}n^{-s}\right)=\sum (a_{n}+b_{n})n^{-s}\,$

$\left(\sum a_{n}n^{-s}\right)\cdot \left(\sum b_{n}n^{-s}\right)=\sum _{n=1}^{\infty }\left(\sum _{d\vert n}^{n}a_{d}b_{n/d}\right)n^{-s}.\,$

and these purely algebraic definitions are consistent with the values achieved within the region of convergence: the multiplication formula is known as Dirichlet convolution.

Formal Dirichlet series

Let R be any ring: an important special case is the ring of integers. A formal power series over R, with variable S is a formal sum $\sum a_{n}n^{-S}$ with coefficients $a_{n}\in R$. Addition and multiplication are now defined purely formally, with no questions of convergence, by the formulae above. The formal Dirichlet series form a ring.