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Difference between revisions of "Hurwitz zeta function"

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(Start article: Hurwitz zeta function)
 
 
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==References==
 
==References==
* Tom M. Apostol, "Introduction to Analytic Number Theory", Undergraduate Texts in Mathematics, Springer (1976) ISBN 0-387-90163-9 {{ZBL|0335.10001}}
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* Tom M. Apostol, "Introduction to Analytic Number Theory", Undergraduate Texts in Mathematics, Springer (1976) {{ISBN|0-387-90163-9}} {{ZBL|0335.10001}}
  
 
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Latest revision as of 09:28, 19 March 2023

generalised zeta function

An Dirichlet series related to the Riemann zeta function which may be used to exhibit properties of various Dirichlet L-functions.

The Hurwitz zeta function $\zeta(\alpha,s)$ is defined for real $\alpha$, $0 < \alpha \le 1$ as $$ \zeta(\alpha,s) = \sum_{n=0}^\infty (n+\alpha)^{-s} \ . $$ The series is convergent, and defines an analytic function, for $\Re s > 1$. The function possesses an analytic continuation to the whole $s$-plane except for a simple pole of residue 1 at $s=1$.

References

  • Tom M. Apostol, "Introduction to Analytic Number Theory", Undergraduate Texts in Mathematics, Springer (1976) ISBN 0-387-90163-9 Zbl 0335.10001
How to Cite This Entry:
Hurwitz zeta function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hurwitz_zeta_function&oldid=52961