Dirichlet series

English

Alternative forms

• Template:L

Etymology

Template:Named-after.

Noun

1. Template:Lb Any infinite series of the form $\sum _{n=1}^{\mathrm{\infty }}\frac{a_n}{n^s}$, where $s$ and each $a_n$ are complex numbers.
   • Template:Quote-book
   • Template:Quote-book
   • 2014, Marius Overholt, A Course in Analytic Number Theory, Template:W, page 157,

     The sum

     $A(s)=\sum _{n=1}^{\mathrm{\infty }}{a}_n{n}^{-s}$

     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 Template:M.
  • Setting $a_n=1$ yields $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}$, which is the formula of the Riemann zeta function.
• When rendered as $\sum _{n=1}^{\mathrm{\infty }}{a}_n{e}^{-{\lambda }_{n}s}$ (for some sequence $\{{\lambda }_n\}$ for which $\{|{\lambda }_n|\}$ increases monotonically), may be called the Template:M.
  • 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 Template:W.

Synonyms

• Template:Sense Template:L, Template:L

Related terms

• Template:L

Translations

• Italian: Template:T
• Polish: Template:T

See also

• Template:L Template:Qualifier
• Template:L
• Template:L

Further reading

• Template:Pedia
• Dirichlet series on Template:W
• Dirichlet L-Series on Template:W
• Dirichlet Series on Template:W