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Distribution of the partition function modulo $m$. (English) Zbl 0984.11050

Let $p\left(n\right)$ denote the usual partition function; $p\left(n\right)$ 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:

$\begin{array}{cc}\hfill p\left(5n+4\right)& \equiv 0\phantom{\rule{10.0pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}5\right),\hfill \\ \hfill p\left(7n+5\right)& \equiv 0\phantom{\rule{10.0pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}7\right),\hfill \\ \hfill p\left(11n+6\right)& \equiv 0\phantom{\rule{10.0pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}11\right)·\hfill \end{array}$

Ramanujan conjectured, and in some cases proved, extensions of these congruences to arbitrary powers of 5, 7, and 11.

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 $m\le 31$. Therefore, the fundamental question of the arithmetic behavior of $p\left(n\right)$ modulo other primes remained entirely a mystery.

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 $m\ge 5$. This fact follows from the author’s first result.

Theorem 1: Let $m\ge 5$ be prime, and let $k$ be a positive integer. Then a positive proportion of the primes $\ell$ have

$p\left(\frac{{m}^{k}{\ell }^{3}n+1}{24}\right)\equiv 0\phantom{\rule{10.0pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}m\right)$

for every non-negative integer $n$ coprime to $\ell$.

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.

After Theorem 1 the author investigates other arithmetic properties of the partition function. For example, a conjecture of Erdős 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.

Conjecture (Newman): If $m$ is an integer, then in every residue class $r\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}m\right)$ there are infinitely many non-negative integers $n$ such that $p\left(n\right)\equiv r\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}m\right)$.

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 $p\left(n\right)\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}3\right)$ remains almost a complete mystery).

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 $m\ge 5$, the sequence of generating functions

$\sum _{{m}^{k}n\equiv -1\phantom{\rule{10.0pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}24\right)}p\left(\frac{{m}^{k}n+1}{24}\right){q}^{n}\phantom{\rule{10.0pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}m\right)$

is periodic in $k$.

This is an important work that represents a true breakthrough in the subject.

##### MSC:
 11P83 Partitions: congruences and congruential restrictions 11F11 Holomorphic modular forms of integral weight 05A17 Partitions of integers (combinatorics)