The generating function for the Dirichlet series $L_m(s)$

Abstract:

The Dirichlet series $L_m(s)$ are of fundamental importance in number theory. Shanks defined the generalized Euler and class numbers in connection with the Dirichlet series, denoted by $\{s_{m,n}\}_{n\geq 0}$. We obtain a formula for the exponential generating function $s_m(x)$ of $s_{m,n}$, where $m$ is an arbitrary positive integer. In particular, for $m>1$, say, $m=bu^2$, where $b$ is square-free and $u>1$, we show that $s_m(x)$ can be expressed as a linear combination of the four functions $w(b,t)\sec (btx)(\pm \cos ((b-p)tx)\pm \sin (ptx))$, where $p$ is a nonnegative integer not exceeding $b$, $t|u^2$ and $w(b,t)=K_bt/u$ with $K_b$ being a constant depending on $b$. Moreover, the Dirichlet series $L_m(s)$ can be easily computed from the generating function formula for $s_m(x)$. Finally, we show that the main ingredient in the formula for $s_{m,n}$ has a combinatorial interpretation in terms of the $\Lambda$-alternating augmented $m$-signed permutations defined by Ehrenborg and Readdy. More precisely, when $m$ is square-free, this answers a question posed by Shanks concerning a combinatorial interpretation of the numbers $s_{m,n}$. When $m$ is not square-free, say $m=bu^2$, the numbers $K_b^{-1}s_{m,n}$ can be written as a linear combination of the numbers of $\Lambda$-alternating augmented $bt$-signed permutations with integer coefficients, where $t|u^2$.