Abstract
In his book Topics in Analytic Number Theory (1973), Hans Rademacher considered the generating function of integer partitions into at most N N parts and conjectured certain limits for the coefficients of its partial fraction decomposition. We carry out an asymptotic analysis that disproves this conjecture, thus confirming recent observations of Sills and Zeilberger (Journal of Difference Equations and Applications 19 (2013)), who gave strong numerical evidence against the conjecture.
Highlights
In his book Topics in Analytic Number Theory [13], Hans Rademacher gave a partial fraction decomposition of the partition generating function j≥1(1 − xj)−1
We proceed by a saddle point analysis of the integral (5), using the approximation of the integrand provided by Lemma 3
We identify a range of width O(N −39/112), delimited by the points z2(N ) := z0 − ρN −39/112 and z3(N ) := z0 + ρN −39/112
Summary
In his book Topics in Analytic Number Theory [13], Hans Rademacher gave a partial fraction decomposition of the partition generating function j≥1(1 − xj)−1. The latter paper presents a recurrence for C0,1,l(N ); the values computed by it do not seem to show convergence, but rather oscillating and unbounded behavior. It is well known, though, that there are number-theoretic problems where the true asymptotics are numerically visible only for very large values. The period p of the oscillations is roughly 32, as observed by Sills and Zeilberger [15] It is independent of l, as is the exponential growth order bN. We comment on the error term in (3), and on possible future work
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