Series (mathematics)
Dirichlet series

A Dirichlet series is one of the form

∑ n = 1 ∞ a n n s, {\displaystyle \sum _{n=1}^{\infty }{a_{n} \over n^{s}},}

where s is a complex number. For example, if all a are equal to 1, then the Dirichlet series is the Riemann zeta function

ζ (s) = ∑ n = 1 ∞ 1 n s. {\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}.}

Like the zeta function, Dirichlet series in general play an important role in analytic number theory. Generally a Dirichlet series converges if the real part of s is greater than a number called the abscissa of convergence. In many cases, a Dirichlet series can be extended to an analytic function outside the domain of convergence by analytic continuation. For example, the Dirichlet series for the zeta function converges absolutely when Re(s) > 1, but the zeta function can be extended to a holomorphic function defined on C ∖ { 1 } {\displaystyle \mathbb {C} \setminus \{1\}} with a simple pole at 1.

This series can be directly generalized to general Dirichlet series.
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