Dirichlet series

alternative Dirichlet's series

1. countableuncountablemathematicsAny infinite series of the form ∑ₙ₌₁ ᪲(a_n)/(nˢ), where s and each a_n are complex numbers.

   We have now given heuristically a large family of multiple Dirichlet series, one for each simply laced Dynkin diagram.

   2012, Daniel Bump, “Chapter 1: Introduction: Multiple Dirichlet Series”, in Daniel Bump, Solomon Friedberg, Dorian Goldfeld, editors, Multiple Dirichlet Series, L-functions and Automorphic Forms, Springer, page 6

   2 more examples

   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.

   2009, Anatoli Andrianov, Introduction to Siegel Modular Forms and Dirichlet Series, Springer (Birkhäuser), page 137

   2014, Marius Overholt, A Course in Analytic Number Theory, American Mathematical Society, page 157, The sum A(s)=∑ₙ₌₁ ᪲a_nn⁻ˢ of a convergent Dirichlet series is a holomorphic (single-valued analytic) function in the half plane σ>σ_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 σ>σ_c(A) by Proposition 3.3.

   Synonyms: general Dirichlet series, ordinary Dirichlet series