Dirichlet series

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

<math>f(s)=\sum_{n=1}^{\infty} \frac{a_n}{n^s}.<math>

The most famous of Dirichlet series is

<math>\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^s},<math>

Other Dirichlet series are:

<math>\frac{1}{\zeta(s)}=\sum_{n=1}^{\infty} \frac{\mu(n)}{n^s}<math>

where μ(n) is the Möbius function,

<math>\frac{\zeta(s-1)}{\zeta(s)}=\sum_{n=1}^{\infty}

\frac{\varphi(n)}{n^s}<math>

where φ(n) is the totient function, and

<math>\frac{\zeta(s)\zeta(s-a)\zeta(s-b)\zeta(s-a-b)}{\zeta(2s-a-b)}

=\sum_{n=1}^{\infty} \frac{\sigma_a(n)\sigma_b(n)}{n^s}<math>

where σ_a(n) is the divisor function.

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