Dirichlet series
nounEtymology Named after German mathematician Peter Gustav Lejeune Dirichlet (1805–1859).
1
Mathematics, Number theory, Sciences Any infinite series of the form ∑ₙ₌₁ ᪲(a_n)/(nˢ), where s and each a_n are complex numbers.
- 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:
- 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:
1 more example
- 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
- Related terms
- Dirichlet L-series, Dirichlet function, Dirichlet L-function, Riemann zeta function