Meaning of Dirichlet series | Babel Free

Definitions

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

countable, uncountable

Equivalents

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.”

“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 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.”

CEFR level