Rademacher's Infinite Partial Fractions Conjecture Is (Almost Certainly) False

<div class="line" id="line-19"> 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)=&sum;n&ge;0pm(n)=1/(1&minus;x)(1&minus;x2)&ctdot;(1&minus;xm).</div><div class="line" id="line-31"> <br/></div><div class="line" id="line-35"> For any &filig;xed m, it is theoretically straightforward to &filig;nd the partial fraction decomposition of the generating function for pm(n). Rademacher made a beautiful and natural conjecture concerning the limiting behavior of the coe&ffilig;cients in the partial fraction decomposition of fm(x) as m &rarr; &infin;, which was published posthumously in 1973. Little progress had been made on this conjecture until just recently, perhaps in large part due to the di&ffilig;culty of actually calculating Rademacher&rsquo;s coe&ffilig;cients for even moderately large values of m. Zeilberger and I found and implemented a fast algorithm for computing Rademacher&rsquo;s coe&ffilig;cients, and as a result, it now seems quite clear that Rademacher&rsquo;s conjecture is almost certainly false! We present some new theorems and conjectures concerning the behavior of Rademacher&rsquo;s coe&ffilig;cients.</div>