Rademacher's Infinite Partial Fraction Conjecture is ( almost certainly) False

By Andrew V. Sills and Doron Zeilberger


.pdf   .ps   .tex  
[Appeared in Journal of Difference Equations and Applications 19(2013), 680-689]
Written: Oct. 21, 2011.

Last update of this webpage (but not article): April 17, 2012.
(previous updates of this page: March 13, 2012)


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 Package

Important: This article is accompanied by Maple package
  • HANS
    [Added March 13, 2012: This new version of HANS contains a new procedure, E01s. For the record, here is the old version of HANS]

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 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 ≤ 800, into one file, called
    HANSC01,
    in Maple readable format. We named that sequence C01r. For example, C017(597) could be gotten (once you uploaded that file), by typing
    C01r[7][597];

  • An even larger list then above, put all the 10 sequences C01j(N) for 1 ≤ j ≤ 10, for 1 ≤ N ≤ 850, into one file, called
    HANSC01a,
    in Maple readable format. We also named that sequence C01r. For example, C017(597) could be gotten (once you uploaded that file), by typing
    C01r[7][597];

  • If you want even more data, but in floating-point, we put all the 40 sequences C01j(N) for 1 ≤ j ≤ 40, for 1 ≤ N ≤ 1000, into one file, called
    HANSC01f,
    in Maple readable format. We named that sequence C01f. For example, the floating-point approximation of C017(999) could be gotten gotten (once you uploaded that file), by typing
    C01f[7][999];

  • If you want to see the 21 sequences C01(-j)(N) for j from 0 to 20 and 1 ≤ N ≤ 500
    the input yields the output ,
    in Maple readable format. We named that sequence C01Minus. To get C01(-j)(N), simply type, C01Minus[j+1][N]; For example, C01(-7)(456) could be gotten (once you uploaded that file), by typing
    C01Minus[8][456];

  • If you want to see conjectured (appx.) asymptotic expressions for C01l(N) for l between 1 and 15,
    the input gives the output.

  • If you want to see the values, in floating-point, of Chkl(N) for 0 < h < k < 10 , k ≥ 3, with gcd(h,k)=1 and for l between 1 and 10, and N between 1 and 400
    the input gives the output.

  • [Added March 13, 2012]

    If you want to see the first 1500 terms of the sequence C011(N)*(2*N)!,
    the input gives the output.

    [Added April 17, 2012]

    If you want to see the first 2000 terms of the sequence C011(N)*(2*N)!,
    the input gives the output.

  • [Added March 13, 2012]

    If you want to see conjectured (appx.) asymptotic expressions for C011(N) (using 1500 terms rather than 900 as in oHANS10)
    the input gives the output.

    [Added April 17, 2012]

    If you want to see conjectured (appx.) asymptotic expressions for C011(N) (using 2000 terms rather than 1500 as in oHANS13)
    the input gives the output.
    [Note the "shifting of the perihelion!", now the maxima are at 1(mod 32) and the minima at 17 (mod 32)]


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