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≥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=bu², 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)(±cos((b-p)tx)±sin(ptx)), where p is an integer satisfying 0≤p ≤ b, t|u² 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}.

  AMS Classification:  11B68, 05A05

  Keywords:  Dirichlet series, generalized Euler and class number, Λ-alternating augmented m-signed permutation, r-cubical lattice, Springer number.

  Download:   PDF