## Experimental Mathematics

### Asymptotic formulas and generalized Dedekind sums

Gert Almkvist

#### Abstract

We find asymptotic formulas as $n\to\infty$ for the coefficients $a(r\hbox{,}\,n)$ defined by \abovedisplayskip2pt plus 2pt \belowdisplayskip2pt plus 2pt \def\nnu{{\hbox{\mathversion{normal}$\scriptstyle\nu$}}} $$\prod_{\nnu=1}^\infty\,(1-x^\nnu)^{-\nnu^r} =\sum_{n=0}^\infty a(r\hbox{,}\,n)x^n\hbox{.}$$ (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.

#### Article information

Source
Experiment. Math. Volume 7, Issue 4 (1998), 343-359.

Dates
First available in Project Euclid: 14 March 2003

http://projecteuclid.org/euclid.em/1047674152

Mathematical Reviews number (MathSciNet)
MR1678083

Zentralblatt MATH identifier
0922.11083

Subjects
Primary: 11P82: Analytic theory of partitions
Secondary: 11F20: Dedekind eta function, Dedekind sums

#### Citation

Almkvist, Gert. Asymptotic formulas and generalized Dedekind sums. Experiment. Math. 7 (1998), no. 4, 343--359. http://projecteuclid.org/euclid.em/1047674152.