On generalized zeta-functions at negative integers
In his thesis [``On a zeta function associated with Bessel's equation'', Univ. Illinois, Urbana, Illinois (1983)] \textit{J. H. Hawkins} studied the function \(z(\nu,s):=\sum ^{\infty}_{n=1}\lambda _ n(\nu)^{-s}\), where \(\lambda _ n(\nu)\) is the nth zero of \(J_{\nu}\), the Bessel function of the first kind and of order \(\nu\), \(\nu >-1\). He showed among other results that z(\(\nu\),s) represents in s an analytic function in the whole plane except for simple poles at \(s=1,-1,-3,-5,....\) In this note a similar situation is studied. But the only condition which the numbers \(\lambda _ n\) now have to satisfy is an asymptotic behavior for large n instead of the fact being zeros of a solution of a certain differential equation (like the Bessel function). So the following theorem is proved: Let P(x) be a real function defined for \(x\geq 1\) with an asymptotic expansion P(x)\(\sim \sum ^{\infty}_{k=-1}A_ kx^{- k}\), \(A_ k\in {\mathbb{R}}\) with \(A:=A_{-1}>0\). Let \(\lambda _ n:=P(n)\) for \(n=1,2,3,... \). If \(z_ p(s):=\sum ^{\infty}_{n=1}\lambda _ n^{-s}\), \(s\in {\mathbb{C}}\), \(Re(s)>1\), then \(z_ p(s)\) represents an analytic function in the whole complex plane except for a simple pole at \(s=1\) (with residue \(A^{-1})\) and perhaps simple poles at \(s=-1,-2,- 3,... \). The corresponding residues can be computed recursively in terms of the coefficients \(A_ k\). This result also applies to the Riemann and Hurwitz zeta-function. The proof uses classical methods, i.e. Mellin transform and the behavior of \(\sum ^{\infty}_{n-1}\exp (-P(n)t)\) for \(t\to 0^ +\).
generalized zeta-functions at negative integers
scientific article; zbMATH DE number 4021256