The generating function for the Dirichlet series Lm(s)

William Y.C. Chen, Neil J.Y. Fan, and Jeffrey Y.T. Jia

  Abstract:  The Dirichlet series Lm(s) are of fundamental importance in number theory. Shanks defined the generalized Euler and class numbers in connection with the Dirichlet series, denoted by {sm, n}n≥0. We obtain a formula for the exponential generating function sm(x) of sm,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 show that sm(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|u2 and w(b, t) = Kbt/u with Kb being a constant depending on b. Moreover, the Dirichlet series Lm(s) can be easily computed from the generating function formula for sm(x). Finally, we show that the main ingredient in the formula for sm,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 sm,n. When m is not square-free and m = bu2, the numbers Kb-1sm,n can be written as a linear combination of the number of ∧-alternating augmented bt-signed permutations with integer coefficients, where t|u2.

  AMS Classification:  11B68, 05A05

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

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