---
tag: 8MWTEF6462
kind: concept
name: Dirichlet series
msc: [11M41, 30B50]
sources:
  - url: https://en.wikipedia.org/wiki/Dirichlet_series
same_as:
  wikidata: Q620595
  wikipedia: Dirichlet series
---

## Summary

A Dirichlet series is a series $\sum_{n \ge 1} a(n) n^{-s}$, where $a$ is an arithmetic function and $s$ is a complex variable.
It converges in a right half-plane and defines a holomorphic function there.
Multiplying two Dirichlet series corresponds to the [[Dirichlet convolution|037F589D1D]] of their coefficients, which makes them the natural generating functions of multiplicative number theory.
The [[Riemann zeta function|1XWBY4N0VT]] and the [[Dirichlet L-functions|9D2Z543EJQ]] are the main examples.

## Explanations

### L1 human

A power series $\sum a_n z^n$ packs a sequence into a function in a way suited to adding indices.
A Dirichlet series $\sum a(n) n^{-s}$ does the same in a way suited to multiplying indices, since $m^{-s} n^{-s} = (mn)^{-s}$.
For the constant sequence $a(n) = 1$ one obtains $\zeta(s) = 1 + 2^{-s} + 3^{-s} + \cdots$.
Facts about divisors and primes then become facts about products and quotients of such functions.