Ono 2000 distribution partition function modulo m

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corollary_2: Ono's corollary that Erdős's conjecture holds, so every prime divides some value of the partition function, with lower bounds for the number of n up to X with m | p(n) when m is not 3; it bears on the first question of Problem 1106 through a step the paper does not take.

theorem_1: Ono's theorem that for a prime m at least 5 and a positive integer k a positive proportion of the primes l make p((m^k l^3 n + 1)/24) divisible by m for every nonnegative n coprime to l, proved from the cusp-form structure of the generating functions F(m,k;z), the Shimura correspondence and Serre's theorem.

theorem_3: Ono's theorem that for a good prime m at least 5 every residue class modulo m contains p(n) for infinitely many n, with counts >> sqrt(X)/log X for the nonzero classes and >> X for the zero class, and Corollary 4 that this covers every prime m < 1000 except possibly m = 3.

theorem_5: Ono's theorem on Ramanujan cycles: for a prime m at least 5 the values p((m^i n + 1)/24) modulo m repeat in i with a period P(m) after a preperiod N(m), both bounded by 48(m^3 - 2m - 1), uniformly in n.

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K. Ono, Distribution of the partition function modulo m, Ann. of Math. (2) 151 (2000), no. 1, 293--307; arXiv:math/0008140.

The copy read for this card is the arXiv copy math/0008140v1 (17 August 2000), a typeset PDF with a clean text layer whose first page is stamped "Annals of Mathematics, 151 (2000), 293--307" and whose running heads carry the journal pages, so PDF p. $n$ is printed p. $292+n$ (fifteen pages). Provenance: a survey download; the download URL was not recorded, and the copy identifies itself only by its arXiv stamp; 167,080 bytes. The journal typesetting was not compared. Read status: claims checked for Theorems 1, 3 and 5 and Corollaries 2 and 4 (statements read clause by clause on the page images, pp. 294--296); the proofs of Theorems 1, 3 and 5, with Theorem 6, Proposition 7 and Theorem 8 (pp. 297--303), were followed at the level of their steps, the external results they cite taken as stated; Corollaries 9--12 (pp. 304--306) were read as statements. Each result page records its own depth. The file prints only its arXiv stamp and the journal line "Annals of Mathematics, 151 (2000), 293--307", and the Annals edition's own terms were not consulted; the arXiv abstract page's license link points to arXiv's assumed license for its 1991-2003 submissions (https://arxiv.org/abs/math/0008140v1, read 2026-10-02), every other right reserved.

Contents

$p(n)$ is the partition function, with $p(0)=1$ and $p(\alpha)=0$ for $\alpha\notin\mathbb N$.

• Background (pp. 293--294): Ramanujan's congruences modulo 5, 7 and 11 and the Atkin--O'Brien congruence (1) modulo 13; the Erdős--Ivić conjecture that infinitely many primes divide some value of $p(n)$, proved by Schinzel (proof in [E-I]); Erdős's conjecture that every prime $m$ has some $n_m\ge0$ with $p(n_m)\equiv0\pmod m$; Schinzel--Wirsing [Sc-W]: the number of primes $m<X$ for which Erdős's conjecture holds is $\gg\log\log X$.
• Theorem 1 (p. 294): fix a prime $m\ge5$ and an integer $k\ge1$. For a positive proportion of all primes $\ell$, the congruence $p\big((m^k\ell^3n+1)/24\big)\equiv0\pmod m$ holds at every integer $n\ge0$ with $\gcd(n,\ell)=1$.
• Corollary 2 (p. 294): Erdős's conjecture holds, so every prime $m$ divides some $p(n_m)$ with $n_m\ge0$. For a prime $m\ne3$ there is a $c_m>0$ such that, for all large $X$, at least $c_m\sqrt X$ (if $m=2$) or $c_mX$ (if $m\ge5$) integers $n\in[0,X]$ have $m\mid p(n)$. The case $m=2$ rests on Ahlgren [A], Nicolas--Ruzsa--Sárközy [Ni-R-Sa] and Serre [S], the case $m=3$ on $p(3)=3$; the paper notes that it is not known whether $p(n)\equiv0\pmod3$ for infinitely many $n$. Example (2) (p. 295): $p(59^4\cdot13n+111247)\equiv0\pmod{13}$ for every $n\ge0$.
• Theorem 3 and Corollary 4 (p. 295): Newman's conjecture (every residue class modulo $m$ is taken by $p(n)$ infinitely often) holds for every "good" prime $m\ge5$, with $\gg\sqrt X/\log X$ values $n\le X$ in each nonzero class and $\gg X$ in the zero class, and hence for every prime $m<1000$ except possibly $m=3$.
• Theorem 5 (p. 296): for prime $m\ge5$ the sequence $p\big((m^kn+1)/24\big)$ modulo $m$ is eventually periodic in $k$ (the "Ramanujan cycles"), with preperiod and period at most $48(m^3-2m-1)$; Section 4 (pp. 303--306) works out the cycles for $5\le m\le23$, with Corollaries 9--12 for $m=13,17,19,23$, and the examples (4), (5) modulo 23 (p. 296) are instances.
• Method (p. 296): the generating functions $F(m,k;z)$ are reductions modulo $m$ of half-integral weight cusp forms in one of two finite-dimensional spaces (Theorem 8, section 3); Theorems 1 and 3 then follow from the Shimura correspondence and Serre's theorem on Galois representations.

Compiled scope

Section 1 (pp. 293--296) was read on the page images, and sections 2--4 (pp. 297--306), which contain the proofs and the examples for $5\le m\le23$, at the depth stated above; the references occupy pp. 306--307. Result pages: Theorem 1, Corollary 2, Theorem 3 (with Corollary 4) and Theorem 5. Nothing here is independently reviewed.

Bears on. #1106: Corollary 2 (p. 294) states that every prime divides some partition number, and that for each prime $m\ge5$ the number of $n\le X$ with $m\mid p(n)$ is $\gg_mX$. The paper does not mention the problem's $F(n)$; the step that every prime then divides $\prod_{k\le n}p(k)$ for large $n$, so that $F(n)\to\infty$, is drawn on the problem's claim page for this paper. The paper gives no rate for $F(n)$ and says nothing about whether $F(n)>n$.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.