Asymptotic formulas and generalized Dedekind sums. (English) Zbl 0922.11083

Almkvist, Gert

Exp. Math. 7, No. 4, 343-359 (1998).

The author studies the numbers \(a(r,n)\) generated by the infinite product expansion \[ f(x)= \prod^\infty_{\nu= 1}(1- x^\nu)^{-\nu^r}= \sum^\infty_{n= 0}a(r,n) x^n. \] The number of plane partitions of \(n\) is \(a(1, n)\). An asymptotic formula for the coefficients \(a(r, n)\) can be obtained by applying the circle method, which requires a knowledge of the behavior of \(f(x)\) near its singularities at the points \(\exp(2\pi ih/k)\) on the unit circle. The author describes four different heuristic approaches for finding expressions for \(f(- t+ \exp(2\pi ih/k))\) for small positive \(t\).
These expressions contain generalized Dedekind sums related to (but different from) those introduced half a century ago by the reviewer [T. M. Apostol, Duke Math. J. 17, 147-157 (1950; Zbl 0039.03801); Pac. J. Math. 2, 1-9 (1952; Zbl 0047.04502)]. There is also a connection with sums of Dedekind type introduced by the reviewer and T. Vu [J. Number Theory 14, 391-396 (1982; Zbl 0487.10012)].

Reviewer: Tom M.Apostol (Pasadena)

MSC:

11P82	Analytic theory of partitions
11F20	Dedekind eta function, Dedekind sums

Keywords:

plane partitions; asymptotic formula; generalized Dedekind sums

Citations:

Zbl 0039.03801; Zbl 0047.04502; Zbl 0487.10012

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