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.

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

• 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;

• 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;

• 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;

• 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;

• 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.

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

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