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

Permanent link to this document
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.


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