The Generating Function for the Dirichlet Series $L_m(s)$

TL;DR

This paper derives a formula for the exponential generating function of generalized Euler and class numbers linked to Dirichlet series, providing a combinatorial interpretation and practical computation methods.

Contribution

It presents a new explicit formula for the generating function of these numbers and connects them to combinatorial structures, answering a longstanding question.

Findings

Explicit formula for $s_m(x)$ in terms of trigonometric functions

Method to compute $L_m(s)$ from the generating function

Combinatorial interpretation of $s_{m,n}$ in terms of m-signed permutations

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 these 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 prove 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 an integer satisfying $0\leq p\leq b$, $t|u^2$ 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 m-signed permutations defined by Ehrenborg and Readdy. In principle, this answers a question posed by Shanks concerning a combinatorial interpretation for the numbers $s_{m,n}$.