Dirichlet series

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Dirichlet series

Wikipedia

Alternative forms

Dirichlet's series

Etymology

Named after German mathematician Peter Gustav Lejeune Dirichlet.

Noun

Dirichlet series (countable and uncountable, plural Dirichlet series)

(number theory) Any infinite series of the form $\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}$ $\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}$ , where $s$ $s$ and each $a_{n}$ $a_{n}$ are complex numbers.
- 2009, Anatoli Andrianov, Introduction to Siegel Modular Forms and Dirichlet Series‎, Springer (Birkhäuser), page 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.
- 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:
  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
  $A(s)=\sum _{n=1}^{\infty }a_{n}n^{-s}$
  
  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 ordinary Dirichlet series.
- 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 $\sum _{n=1}^{\infty }a_{n}e^{-\lambda _{n}s}$ $\sum _{n=1}^{\infty }a_{n}e^{-\lambda _{n}s}$ (for some sequence $\{\lambda _{n}\}$ $\{\lambda _{n}\}$ for which $\{\vert \lambda _{n}\vert \}$ $\{\vert \lambda _{n}\vert \}$ increases monotonically), may be called the general Dirichlet series.
- 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 Laplace-Stieltjes transform.