On the Laurent coefficients of a class of Dirichlet series
The generalized Euler constants \(\gamma_k(r,m)\) are defined by \[ \gamma_k(a,q) = \lim_{x\to\infty} \bigg\{\sum_{{ 0<n<x}, \;{ n\equiv a\bmod q}} \frac{\log^kn}{n} - \frac{\log^{k+1}x}{q(k+1)}\bigg\}. \] When \(m=1\), set \(\gamma_k(1,1)=\gamma_k\). The generating function of \(\gamma_k\) is the Riemann zeta-function \(\zeta(s)\). We have the Laurent expansion \[ \zeta(s)=\frac{1}{s-1}+\sum^\infty_{k=0} \frac{(-1)^k\gamma_k}{k!} (s-1)^k. \] \textit{S. Kanemitsu} [Théorie des nombres, C. R. Conf. Int., Québec/Can. 1987, 459--474 (1989; Zbl 0682.10027)] posed the problem of finding the generating function in the general case. Answering the problem the author considers the following zeta function \[ \zeta(s; a, q)=\sum_{{n=1}\atop{n\equiv a\bmod q}}^\infty \frac 1{n^s}. \] Clearly \(\zeta(s; a, q)=q^{-s}\zeta(s, a/q),\) where \(\zeta(s, \alpha)\) is the Hurwitz zeta-function. He proves that the Laurent expansion \[ \zeta(s; a, q)=\frac{1}{q(s-1)}+\sum^\infty_{k=0} \frac{(-1)^k}{k!}\gamma_k(a, q) (s-1)^k \] is valid. Using this function the author derives the preceding results of \textit{K. Dilcher} [Math. Comput. 59, 259--282, S21-S24 (1992; Zbl 0752.11033)] and S. Kanemitsu [loc. cit.] on \(\gamma_k(a, q)\) in a unified way.
scientific article; zbMATH DE number 1997678
scientific article