The generating function for the Dirichlet series L_{m}(s)
William Y.C. Chen, Neil J.Y. Fan, and Jeffrey Y.T. Jia
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≥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)(±cos((b - p)tx) ± sin(ptx)), where p is a nonnegative integer not exceeding b, t|u^{2} and w(b, t) = K_{b}t/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 ∧-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 and m = bu^{2}, the numbers K_{b}^{-1}s_{m,n} can be written as a linear combination of the number of ∧-alternating augmented bt-signed permutations with integer coefficients, where t|u^{2}. AMS Classification: 11B68, 05A05 Keywords: Dirichlet series, generalized Euler and class number, ∧-alternating augmented m-signed permutation, r-cubical lattice, Springer number. Download: PDF |