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Statement

Theorem 5 (p. 296). Let $m\ge5$ be prime. There are integers $N(m)$ and $P(m)$ with

$$0\le N(m)\le48(m^3-2m-1),\qquad 1\le P(m)\le48(m^3-2m-1),$$

such that for every $i>N(m)$ and every nonnegative integer $n$,

$$p\left(\frac{m^in+1}{24}\right)\equiv p\left(\frac{m^{P(m)+i}\cdot n+1}{24}\right)\pmod m .$$

Here $p(\alpha)=0$ for $\alpha\notin\mathbb N$ (p. 293). The paper calls the periods of the sequence of generating functions $F(m,k;z)$ of display (3) (p. 295) in $k$ "Ramanujan cycles" (p. 296).

Worked cases (pp. 303--306). For $m=5,7,11$ the cycles are degenerate, $F(m,k;z)\equiv0\pmod m$ for every $k\ge1$, by Ramanujan's congruences. For $m=13,17,19,23$ Corollaries 9, 10, 11 and 12 express $p\big((m^{2k+1}(24n+a)+1)/24\big)$ and $p\big((m^{2k+2}(24n+23)+1)/24\big)$ modulo $m$, for all nonnegative $k$ and $n$, as a constant times $6^k$, $6^k$, $10^k$ or $5^k$ respectively, times the $n$-th coefficient of a power of $\prod_{n\ge1}(1-q^n)$ multiplied by $1$, $E_4$, $E_6$ or $E_4E_6$ respectively, with $a=11,7,5,1$. Displays (4) and (5) (p. 296) are instances for $m=23$, (4) from Corollary 12 at $n=0$.

Source. K. Ono, Distribution of the partition function modulo $m$, Ann. of Math. (2) 151 (2000), no. 1, 293--307; Theorem 5 on p. 296, its proof on p. 303, the examples of Section 4 on pp. 303--306. Pages are the journal's, as printed in the running heads of the copy identified on the source card.

Read depth. Claims checked: Theorem 5 was read clause by clause on the page image, and Corollaries 9--12 were read as statements. The proof was read; the dimension bounds it cites were not checked, and the Sturm-bound computations behind Section 4 were not redone. Nothing here is independently reviewed.

Proof pointer

Page 303. By Theorem 8 each $F(m,k;z)$ lies in one of two finite-dimensional $\mathbb F_m$-vector spaces (the character $\chi\chi_m^{k-1}$ depends only on the parity of $k$); Proposition 7 makes the sequence an orbit of $U(m)$, so it is eventually periodic, and upper bounds for the dimensions of spaces of cusp forms (Cohen and Oesterlé, the paper's [C-O]) give the bound $48(m^3-2m-1)$. Display (3) and Theorem 6 translate the periodicity into the congruence.

Dependencies

Theorem 6, Proposition 7 and Theorem 8 of the same paper; H. Cohen and J. Oesterlé, Dimensions des espaces de formes modulaires, Lecture Notes in Math. 627 (1977), 69--78 (the paper's [C-O]).

Bears on

No Erdős problem in this corpus.