A Dirichlet series is a series of the formMultiplication of absolutely convergent Dirichlet series corresponds to Dirichlet convolution of their coefficients.
An Euler product factors a Dirichlet series into local factors indexed by prime numbers. For a multiplicative arithmetic function and in a half-plane of absolute convergence,
For every complex number with ,Applying this to each prime-power term in logarithms of Euler products proves thatfor and every completely multiplicative bounded by one.
Perron's formula recovers a summatory arithmetic function from its Dirichlet series by the inverse Mellin integralwith the usual convergence and endpoint conventions.
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A Dirichlet series is a type of infinite series of the form: \[ D(s) = \sum_{n=1}^\infty \frac{a_n}{n^s} \] where \( s \) is a complex variable, \( a_n \) are complex coefficients, and \( n \) ranges over the positive integers. The series converges for certain values of the complex variable \( s \) depending on the properties of the coefficients \( a_n \).