Rademacher's Infinite Partial Fraction Conjecture is False

By Andrew V. Sills and Doron Zeilberger

.pdf   .ps   .tex  
Written: Oct. 8, 2011.
The first-named author's "academic grandfather", Hans Rademacher, was a great number theorist, but even great mathematicians sometimes make false conjectures. In this article we prove (empirically) that a conjecture made by Rademacher in his posthumously published classic "Topics in Number Theory" is (very!) false as stated, but if you replace "infinity" by some good-old finite numbers it may be resurrected.

Maple Packages

Important: This article is accompanied by Maple packages

Sample Input and Output for HANS

  • If you want to see the first 700 terms of the sequence C011(N) as exact rational numbers, followed by their floating-point renditions, that overwhelmingly shatter Rademacher's conjecture by showing that that sequence does not converge to anything (in particular not to -0.29292754...) but instead eventually oscillates widely getting ever-so-clse to plus infinity and ever-so-close to negative infinity (with a period that seems to be 32), the input gives the output.

  • If you want to see the first 800 terms of the sequence C011(N) in their floating-point-approximation renditions, that even more overwhelmingly shatter Rademacher's conjecture by showing that that sequence does not converge to anything (in particular not to -0.29292754...) but instead eventually oscillates widely getting ever-so-clse to plus infinity and ever-so-close to negative infinity (with a period that seems to be 32), the input gives the output.

  • If you want to see the first 500 terms of the sequences C01j(N) for j from 1 to 10, both in exact rational arithmetic, and in floating-point,
    the input gives the output.

  • If you want to see the first 700 terms of the sequence C121(N)
    the input gives the output.

  • If you want to see the "closest encounters" of the sequences Chkl(N) to Radmacher's alleged (but wrong!) "limit" (that he called, with wishful thinking, Chkl(∞)) for 0 ≤ h < k ≤ 3, (gcd(h,k)=1), l ≤5, and N ≤ 250,
    the input gives the output.

  • If you want to conduct your own computer experiments with our data, we have put all the 10 sequences C01j(N) for 1 ≤ j ≤ 10, for 1 ≤ N ≤ 700, into one file, called
    HANS700,
    in Maple readable format. We named that sequence HANS700. For example, C017(597) could be gotten by typing (once you uploaded that file), by typing
    HANS700[7][597];

Doron Zeilberger's List of Papers

Doron Zeilberger's Home Page