Dirichlet series - Weblio 英和・和英辞典 (original) (raw)

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Dirichlet series

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Dirichlet series

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Dirichlet series

名詞

Dirichlet series (countable かつ uncountable, 複数形 **Dirichlet series)

(number theory) Any infinite series of the form ∑ n = 1 ∞ a n n s {\displaystyle \sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}} , where s {\displaystyle s} and each a n {\displaystyle 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, 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 ) = ∑ n = 1 ∞ a n n − s {\displaystyle A(s)=\sum _{n=1}^{\infty }a_{n}n^{-s}} $A(s)=\sum _{n=1}^{\infty }a_{n}n^{-s}$
  of a convergent Dirichlet series is a holomorphic (single-valued analytic) function in the half plane σ > σ c ( A ) {\displaystyle \sigma >\sigma _{c}(A)} $\sigma >\sigma _{c}(A)$ , and the terms of the Dirichlet series are holomorphic in the whole complex plane, and the series converges uniformly on every compact subset of σ > σ c ( A ) {\displaystyle \sigma >\sigma _{c}(A)} $\sigma >\sigma _{c}(A)$ by Proposition 3.3.

Dirichlet series - Weblio 英和・和英辞典 (original) (raw)

Dirichlet series

Dirichlet series

Dirichlet series

Dirichlet series

Dirichlet series

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