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Zbl 0984.11050
Ono, Ken
Distribution of the partition function modulo $m$. (English)
[J] Ann. Math. (2) 151, No.1, 293-307 (2000). ISSN 0003-486X; ISSN 1939-0980/e

Let $p(n)$ denote the usual partition function; $p(n)$ is the number of ways to write the natural number $n$ as the sum of a non-increasing sequence of positive integers. The arithmetic properties of this function have been the subject of much study over the eighty years since Ramanujan proved the following famous congruences: $$\aligned p(5n+4)&\equiv 0\pmod 5,\\ p(7n+5)&\equiv 0\pmod 7,\\ p(11n+6)&\equiv 0\pmod {11}.\endaligned$$ Ramanujan conjectured, and in some cases proved, extensions of these congruences to arbitrary powers of $5$, $7$, and $11$. \par Through the end of the 1960s, further congruences (and related arithmetic phenomena) were discovered by Atkin, Newman, O'Brien, and Swinnerton-Dyer. These works involve only primes $m$ with $m\leq 31$. Therefore, the fundamental question of the arithmetic behavior of $p(n)$ modulo other primes remained entirely a mystery. \par This paper represents a true breakthrough with regards to this question. In particular, the author proves the remarkable result that infinitely many congruences like Ramanujan's exist modulo $m$ for every prime $m\geq 5$. This fact follows from the author's first result. \par Theorem 1: Let $m\geq 5$ be prime, and let $k$ be a positive integer. Then a positive proportion of the primes $\ell$ have $$p\fracwithdelims(){m^k\ell^3n+1}{24}\equiv 0\pmod m$$ for every non-negative integer $n$ coprime to $\ell$. \par The proof of this result employs heavily the theories of integral and half-integral weight modular forms. First, the author constructs certain half-integral weight cusp forms whose Fourier coefficients interpolate values of the partition function modulo primes $m$. He employs the Shimura correspondence to lift these cusp forms into integral weight spaces; in these spaces he applies the theory of modular Galois representations (as developed by Deligne and Serre) in order to obtain the result. Thus congruences like Ramanujan's may be viewed as ``footprints" of such deep objects as modular forms and Galois representations. \par After Theorem 1 the author investigates other arithmetic properties of the partition function. For example, a conjecture of Erd\H{o}s was that every prime $m$ divides some value of the partition function; an easy corollary of Theorem~1 gives a much stronger statement in this direction. The author also considers the following old conjecture of Newman. \par Conjecture (Newman): If $m$ is an integer, then in every residue class $r\pmod m$ there are infinitely many non-negative integers $n$ such that $p(n)\equiv r\pmod m$. \par In Theorem 3, the author proves that if $m$ is a ``good'' prime, then Newman's conjecture is true for $m$. As a corollary, he proves that Newman's conjecture is true for every prime $m<1000$, with the possible exception of $m=3$ (indeed, the behavior of $p(n)\pmod 3$ remains almost a complete mystery). \par Finally, in Theorem~5 the author develops the theory of ``Ramanujan cycles''; these are periodic relations which arise from his proof that, for each prime $m\geq 5$, the sequence of generating functions $$\sum_{m^kn\equiv {-1}\pmod {24}}p\fracwithdelims(){m^kn+1}{24}q^n\pmod m$$ is periodic in $k$. \par This is an important work that represents a true breakthrough in the subject.
[Scott Ahlgren (Urbana)]

MSC 2000:: *11P83 Partitions: congruences and congruential restrictions
11F11 Modular forms, one variable
05A17 Partitions of integres (combinatorics)

Keywords: partitions; congruences; distribution of the partition function; Ramanujan-like congruences; half-integral weight cusp forms; conjecture of Erd\H{o}s; Newman's conjecture; Ramanujan cycles

Cited in: Zbl 0986.11071 Zbl 1007.11061

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