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In mathematics, a Dirichlet series is any series of the form $$ \sum_{n=1}^{\infty} \frac{a_n}{n^s}, $$ where $s$ and an are complex numbers and $n = 1, 2, 3, \dots$ . It is a special case of general Dirichlet series.

Dirichlet series play a variety of important roles in analytic number theory. The most usually seen definition of the Riemann $\zeta$ function is a Dirichlet series, as are the Dirichlet $L$-functions. It is conjectured that the Selberg class of series obeys the generalized Riemann hypothesis. The series is named in honor of Johann Peter Gustav Lejeune Dirichlet.

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