Dirichlet series

Etymology
.

Noun

 * 1)  Any infinite series of the form $$\sum_{n=1}^\infty \frac{a_n}{n^s}$$, where $$s$$ and each $$a_n$$ are complex numbers.
 * 2) * 2014, Marius Overholt, A Course in Analytic Number Theory,, page 157,
 * The sum
 * $$A(s)=\sum_{n=1}^\infty a_n n^{-s}$$
 * of a convergent Dirichlet series is a holomorphic (single-valued analytic) function in the half plane $$\sigma>\sigma_c(A)$$, and the terms of the Dirichlet series are holomorphic in the whole complex plane, and the series converges uniformly on every compact subset of $$\sigma>\sigma_c(A)$$ by Proposition 3.3.
 * $$A(s)=\sum_{n=1}^\infty a_n n^{-s}$$
 * of a convergent Dirichlet series is a holomorphic (single-valued analytic) function in the half plane $$\sigma>\sigma_c(A)$$, and the terms of the Dirichlet series are holomorphic in the whole complex plane, and the series converges uniformly on every compact subset of $$\sigma>\sigma_c(A)$$ by Proposition 3.3.

Usage notes

 * In the above form, sometimes called the.
 * Setting $$a_n = 1$$ yields $$\sum_{n=1}^\infty \frac{1}{n^s}$$, which is the formula of the Riemann zeta function.
 * When rendered as $$\sum_{n=1}^\infty a_n e^{-\lambda_n s}$$ (for some sequence $$\{\lambda_n\}$$ for which $$\{\vert\lambda_n\vert\}$$ increases monotonically), may be called the.
 * This reduces to the ordinary form if $$\lambda_n=\ln n$$.
 * It is in the general form that the series is most plainly seen to be a special case of the.

Related terms

 * Dirichlet L-series

Translations

 * Italian: serie di Dirichlet