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

Andrew Sills; Doron Zeilberger

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

Andrew Sills, Doron Zeilberger

Mathematical Sciences

Rutgers, The State University of New Jersey

Research output: Contribution to conference › Presentation

Abstract

<div class="line" id="line-5"> This is joint work with Doron Zeilberger. Let p m ( n ) denote the number of partitions of n with at most m summands. The generating function of p m ( n ) is</div><div class="line" id="line-226" style="text-align: center;"> f m ( x )= ∑ n ≥0 p m ( n ) x n =1/(1− x )(1− x 2 )&ctdot;(1− x m ). </div><div class="line" id="line-833" style="text-align: center;"> </div><div class="line" id="line-576"> For any fixed m , it is theoretically straightforward to find the partial fraction decomposition of the generating function for p m ( n ). Rademacher made a beautiful and natural conjecture concerning the limiting behavior of the coefficients in the partial fraction decomposition of f m ( 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 of the data we collected, it now seems quite clear that Rademacher's conjecture must be FALSE! We present some new theorems and conjectures concerning the behavior of Rademacher's coefficients.</div>

Original language	American English
State	Published - Mar 22 2012
Event	University of Florida Number Theory Seminar - Gainsville, FL Duration: Mar 22 2012 → …

Conference

Conference	University of Florida Number Theory Seminar
Period	03/22/12 → …

Keywords

Infinite partial fractions
Number theory
Rademacher

DC Disciplines

Mathematics

Cite this

@conference{a6cd5d8ca5864cfcb7efbc88bcdde02c,

title = "Rademacher{\textquoteright}s Inﬁnite Partial Fractions Conjecture Is (Almost Certainly) False",

abstract = " This is joint work with Doron Zeilberger. Let p m ( n ) denote the number of partitions of n with at most m summands. The generating function of p m ( n ) is f m ( x )= ∑ n ≥0 p m ( n ) x n =1/(1− x )(1− x 2 )&ctdot;(1− x m ). For any fixed m , it is theoretically straightforward to find the partial fraction decomposition of the generating function for p m ( n ). Rademacher made a beautiful and natural conjecture concerning the limiting behavior of the coefficients in the partial fraction decomposition of f m ( 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 of the data we collected, it now seems quite clear that Rademacher's conjecture must be FALSE! We present some new theorems and conjectures concerning the behavior of Rademacher's coefficients.",

keywords = "Infinite partial fractions, Number theory, Rademacher",

author = "Andrew Sills and Doron Zeilberger",

year = "2012",

month = mar,

day = "22",

language = "American English",

note = "University of Florida Number Theory Seminar ; Conference date: 22-03-2012",

}

TY - CONF

T1 - Rademacher’s Inﬁnite Partial Fractions Conjecture Is (Almost Certainly) False

AU - Sills, Andrew

AU - Zeilberger, Doron

PY - 2012/3/22

Y1 - 2012/3/22

N2 - This is joint work with Doron Zeilberger. Let p m ( n ) denote the number of partitions of n with at most m summands. The generating function of p m ( n ) is f m ( x )= ∑ n ≥0 p m ( n ) x n =1/(1− x )(1− x 2 )&ctdot;(1− x m ). For any fixed m , it is theoretically straightforward to find the partial fraction decomposition of the generating function for p m ( n ). Rademacher made a beautiful and natural conjecture concerning the limiting behavior of the coefficients in the partial fraction decomposition of f m ( 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 of the data we collected, it now seems quite clear that Rademacher's conjecture must be FALSE! We present some new theorems and conjectures concerning the behavior of Rademacher's coefficients.

AB - This is joint work with Doron Zeilberger. Let p m ( n ) denote the number of partitions of n with at most m summands. The generating function of p m ( n ) is f m ( x )= ∑ n ≥0 p m ( n ) x n =1/(1− x )(1− x 2 )&ctdot;(1− x m ). For any fixed m , it is theoretically straightforward to find the partial fraction decomposition of the generating function for p m ( n ). Rademacher made a beautiful and natural conjecture concerning the limiting behavior of the coefficients in the partial fraction decomposition of f m ( 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 of the data we collected, it now seems quite clear that Rademacher's conjecture must be FALSE! We present some new theorems and conjectures concerning the behavior of Rademacher's coefficients.

KW - Infinite partial fractions

KW - Number theory

KW - Rademacher

M3 - Presentation

T2 - University of Florida Number Theory Seminar

Y2 - 22 March 2012

ER -

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

Abstract

Conference

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