Rademacher's Infinite Partial Fractions Conjecture Is (Almost Certainly) False
A partition of n is a representation of n as a sum of positive integers where the order of summands is considered irrelevant. Let pm(n) denote the number of partitions of n with at most m summands. The generating function of pm(n) is fm(x)=∑n≥0pm(n)=1/(1−x)(1−x2)⋯(1−xm). For any ﬁxed m, it is theoretically straightforward to ﬁnd the partial fraction decomposition of the generating function for pm(n). Rademacher made a beautiful and natural conjecture concerning the limiting behavior of the coeﬃcients in the partial fraction decomposition of fm(x) as m → ∞, which was published posthumously in 1973. Little progress had been made on this conjecture until just recently, perhaps in large part due to the diﬃculty of actually calculating Rademacher’s coeﬃcients for even moderately large values of m. Zeilberger and I found and implemented a fast algorithm for computing Rademacher’s coeﬃcients, and as a result, it now seems quite clear that Rademacher’s conjecture is almost certainly false! We present some new theorems and conjectures concerning the behavior of Rademacher’s coeﬃcients.
Joint Mathematics Meetings (JMM)
San Diego, CA
Sills, Andrew V., Doron Zeilberger.
"Rademacher's Infinite Partial Fractions Conjecture Is (Almost Certainly) False."
Mathematical Sciences Faculty Presentations.