Dirichlet series

English Alternative forms en Dirichlet's series Etymology en German mathematician Peter Gustav Lejeune Dirichlet 1805 1859 =. Noun ~ * en number theory Any infinite series of the form , where and each are complex numbers. en 2009 Anatoli Andrianov Introduction to Siegel Modular Forms and Dirichlet Series Springer (Birkhäuser) https://books.google.com.au/books?id=kUoZVq4SXMIC&pg=PA137&dq=%22Dirichlet+series%22&hl=en&sa=X&ved=0ahUKEwjMm_ypsezdAhULg5AKHS-JAKwQ6AEIdzAN#v=onepage&q=%22Dirichlet%20series%22&f=false 137 Traditionally, starting from Euler, multiplicativity of arithmetic sequences is customarily expressed in the form of an Euler product factorization of the generating Dirichlet series. It turns out that in the situation of modular forms, suitable Dirichlet series constructed by Fourier coefficients of eigenfunctions of Hecke operators can be expressed through Dirichlet series formed by the corresponding eigenvalues. en 2012 Daniel Bump Chapter 1: Introduction: Multiple Dirichlet Series Daniel Bump; Solomon Friedberg; Dorian Goldfeld Multiple Dirichlet Series, L-functions and Automorphic Forms Springer https://books.google.com.au/books?id=ajh2PeQn_AkC&pg=PA6&dq=%22Dirichlet+series%22&hl=en&sa=X&ved=0ahUKEwjMm_ypsezdAhULg5AKHS-JAKwQ6AEIfTAO#v=onepage&q=%22Dirichlet%20series%22&f=false 6 We have now given heuristically a large family of multiple Dirichlet series, one for each simply laced Dynkin diagram. 2014, Marius Overholt, A Course in Analytic Number Theory, American Mathematical Society, page 157, The sum of a convergent Dirichlet series is a holomorphic (single-valued analytic) function in the half plane , and the terms of the Dirichlet series are holomorphic in the whole complex plane, and the series converges uniformly on every compact subset of by Proposition 3.3. Usage notes In the above form, sometimes called the en ordinary Dirichlet series. Setting yields , which is the formula of the Riemann zeta function. When rendered as (for some sequence for which increases monotonically), may be called the en general Dirichlet series. This reduces to the ordinary form if . It is in the general form that the series is most plainly seen to be a special case of the Laplace-Stieltjes transform. Synonyms infinite series en general Dirichlet series, en ordinary Dirichlet series Related terms en Dirichlet L-series Dirichlet L-series Translations infinite series Italian: it serie di Dirichlet f Polish: pl szereg Dirichleta m See also en Dirichlet function unrelated concept en Dirichlet L-function Dirichlet L-function en Riemann zeta function Further reading Riemann zeta function Dirichlet series on Encyclopedia of Mathematics Dirichlet L-Series on MathWorld Wolfram MathWorld Dirichlet Series on MathWorld Wolfram MathWorld