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Some results on the generalized Stieltjes constants. (English) Zbl 0808.11054

For \(0<a\leq 1\) and \(\sigma>1\) the Hurwitz zeta function \(\zeta(s,a)\) is defined by \(\zeta(s,a)= \sum_{n=0}^ \infty (n+a)^{-s}\), \(s=\sigma+ it\). \(\zeta(s,a)\) has the Laurent series expansion \[ \zeta(s,a)= {1\over {s-1}}+ \sum_{n=0}^ \infty {{(-1)^ n \nu_ n(a)} \over {n!}} (s- 1)^ n \] where the \(\gamma_ n(a)\) are known as generalized Stieltjes constants. The author proves a number of expressions and estimates for these constants.

MSC:

11M35 Hurwitz and Lerch zeta functions
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