DirichletL-function and power series for Hurwitz zeta function
Copyright policy )DirichletL-function and power series for Hurwitz zeta function
Let \(\zeta(s,a)= \sum^ \infty_{n=0} (n+a)^{-s}\) be the Hurwitz zeta-function. For fixed complex \(s\neq 1\) it is shown that \(\zeta(s,a)- a^{-s}\) is holomorphic, as a function of \(a\), in the disc \(| a|<1\), with power series expansion \(\zeta(s,a)=\sum^ \infty_{n=0} {{-s} \choose n} \zeta(s+n)a^ n\). As corollaries one gets expansions for \(\sum_{\pmod q} L(s,\chi)^ 2\), where \(L(s,\chi)\) are the Dirichlet \(L\)-functions, similar to those of \textit{M. Katsurada} and \textit{K. Matsumoto} [Math. Z. 208, 23-39 (1991; Zbl 0744.11041)]. In particular, for \(q\) prime and \(s\) fixed, one gets an asymptotic expansion in descending powers of \(q\), generalizing the result for \(s=1/2\) obtained by the reviewer [Comment. Math. Helv. 56, 148-161 (1981; Zbl 0457.10020)].
- Shanghai Institute for Science of Science China (People's Republic of)
power series expansion, Hurwitz zeta-function, asymptotic expansion, mean-value, \(\zeta (s)\) and \(L(s, \chi)\), Dirichlet \(L\)- functions, Hurwitz and Lerch zeta functions
power series expansion, Hurwitz zeta-function, asymptotic expansion, mean-value, \(\zeta (s)\) and \(L(s, \chi)\), Dirichlet \(L\)- functions, Hurwitz and Lerch zeta functions
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