Approximating and bounding fractional Stieltjes constants

Abstract

We discuss evaluating fractional Stieltjes constants γα(a), arising naturally from the Laurent series expansions of the fractional derivatives of the Hurwitz zeta functions ζ(α)(s, a). We give an upper bound for the absolute value of Cα(a) = γα(a) − log(a)/a and an asymptotic formula C̃α(a) for Cα(a) that yields a good approximation even for most small values of α. We bound |C̃α(a)| and based on this conjecture a tighter bound for |Cα(a)|