Hurwitz zeta function

In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables $s$ with $Re(s) > 1$ and $a \neq 0, -1, -2, \dots$ by

\zeta (s,a)=\sum _{n=0}^{\infty }{\frac {1}{(n+a)^{s}}}.

This series is absolutely convergent for the given values of $s$ and $a$ and can be extended to a meromorphic function defined for all $s \neq 1$ . The Riemann zeta function is $ζ(s,1)$ . The Hurwitz zeta function is named after Adolf Hurwitz, who introduced it in 1882.^[1]

Hurwitz zeta function corresponding to $a = 1/3$ . It is generated as a Matplotlib plot using a version of the Domain coloring method.^[2]

Hurwitz zeta function corresponding to $a = 24/25$ .

Hurwitz zeta function as a function of $a$ with $s = 3 + 4 i$ .

^ Hurwitz, Adolf (1882). "Einige Eigenschaften der Dirichlet'schen Functionen ${\textstyle F(s)=\sum \left({\frac {D}{n}}\right)\cdot {\frac {1}{n}}}$ , die bei der Bestimmung der Classenanzahlen binärer quadratischer Formen auftreten". Zeitschrift für Mathematik und Physik (in German). 27: 86–101.
^ "Jupyter Notebook Viewer".

[1] Hurwitz, Adolf (1882). "Einige Eigenschaften der Dirichlet'schen Functionen ${\textstyle F(s)=\sum \left({\frac {D}{n}}\right)\cdot {\frac {1}{n}}}$ , die bei der Bestimmung der Classenanzahlen binärer quadratischer Formen auftreten". Zeitschrift für Mathematik und Physik (in German). 27: 86–101.

[2] "Jupyter Notebook Viewer".

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