Stieltjes constants for Hurwitz-type L-series attached to periodic arithmetic functions

We investigate generalized Stieltjes constants attached to Hurwitz-type Dirichlet series of the form $$L(s,f,x)=\sum _{n\ge 0}\frac{f(n)}{(n+x{)}^s}$$ where $f$ is a periodic arithmetic function and $x\in \mathbb{C}\mathrm{\backslash }\{0,-1,-2,\dots \}\mathrm{.}$ We prove that these series admit meromorphic continuation to the whole complex plane, with at most a simple pole at $s=1$. We further derive functional equations satisfied by $L(s,f,x)\mathrm{.}$ Our main result gives an explicit formulas of the Stieltjes constants ${\gamma }_n(f,x)$ appearing in the Laurent expansion at $s=1$. In particular, we obtain explicit evaluations in the case of rational $x$. As an application, we derive explicit closed-form expressions for the first three generalized Stieltjes constants associated with nonprincipal real primitive odd Dirichlet characters arising from quadratic fields. These expressions are given in terms of the class number of the corresponding quadratic number field. These results extend and unify recent work of Blagouchine and others on generalized Stieltjes constants and Hurwitz-type $L$-functions.