ASYMPTOTIC FORMULAS AND GENERALIZED DEDEKIND SUMS

GERT ALMKVIST

Abstract. We find asymptotic formulas as n → ∞ for the coefficients a(r, n) defined by

∏_{ν=1}^{∞} (1 − x^{ν})^{−ν^{r}} = ∑_{n=0}^{∞} a(r, n)x^{n}.

(The case r = 1 gives the number of plane partitions of n.) Generalized Dedekind sums occur naturally and are studied using the Finite Fourier Transform. The methods used are unorthodox; many of the computations are not justified but the result is in many cases very good numerically. The last section gives various formulas for Kinkelin’s constant.