Dirichlet series can be used as generating series for counting weighted sets of objects with respect to a weight which is combined multiplicatively when taking Cartesian products.
Suppose that A is a … See more

A formal Dirichlet series over a ring R is associated to a function a from the positive integers to R See more

Given
$${\displaystyle F(s)=\sum _{n=1}^{\infty }{\frac {f(n)}{n^{s}}}}$$
it is possible to show that
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The inverse Mellin transform of a Dirichlet series, divided by s, is given by Perron's formula. Additionally, if See more

The most famous example of a Dirichlet series is
$${\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}},}$$
whose analytic … See more

Given a sequence $${\displaystyle \{a_{n}\}_{n\in \mathbb {N} }}$$ of complex numbers we try to consider the value of
$${\displaystyle f(s)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}}$$
as a function of the complex variable s. In order for this to … See more

Suppose
$${\displaystyle F(s)=\sum _{n=1}^{\infty }f(n)n^{-s}}$$
and
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