Approximating and bounding fractional Stieltjes constants

Abstract

We discuss evaluating fractional Stieltjes constants $\gamma_{\alpha}(a)$, arising naturally from the Laurent series expansions of the fractional derivatives of the Hurwitz zeta functions $\zeta^{(\alpha)}(s,a)$. We give an upper bound for the absolute value of $C_\alpha(a)=\gamma_\alpha(a)-\log^\alpha(a)/a$ and an asymptotic formula $\widetilde{C}_{\alpha}(a)$ for $C_{\alpha}(a)$ that yields a good approximation even for most small values of $\alpha$. We bound $|\widetilde{C}_{\alpha}(a)|$ and based on this we conjecture a tighter bound for $|C_\alpha(a)|$.