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Statement

Conjecture (M. Newman) (p. 295, quoted). "If $m$ is an integer, then for every residue class $r\pmod m$ there are infinitely many nonnegative integers $n$ for which $p(n)\equiv r\pmod m$."

Good primes (p. 295). A prime $m\ge5$ is good if for every residue class $r\pmod m$ there is a nonnegative integer $n_r$ with $mn_r\equiv-1\pmod{24}$ and $p\big((mn_r+1)/24\big)\equiv r\pmod m$.

Theorem 3 (p. 295). Let $m\ge5$ be a good prime. Then Newman's conjecture holds for $m$, and for each residue class $r\pmod m$,

$$\#\{0\le n\le X\ :\ p(n)\equiv r\pmod m\}\gg_{r,m} \begin{cases}\sqrt X/\log X & \text{if } 1\le r\le m-1,\\ X & \text{if } r=0.\end{cases}$$

Corollary 4 (p. 295, quoted). "Newman's conjecture is true for every prime $m<1000$ with the possible exception of $m=3$." The paper presents it as the outcome of a computation of good primes, run with code written by J. Haglund and C. Haynal, and does not list the computation's output. It records (p. 295) that Atkin, Newman and Kolberg had verified the conjecture for $m=2,5,7,11$ and $13$, adding that the case $m=11$ is not proved in those papers but follows by an easy modification of their arguments.

Source. K. Ono, Distribution of the partition function modulo $m$, Ann. of Math. (2) 151 (2000), no. 1, 293--307; the definition, Theorem 3 and Corollary 4 on p. 295, the proof of Theorem 3 on pp. 302--303. Pages are the journal's, as printed in the running heads of the copy identified on the source card.

Read depth. Claims checked: the definition, Theorem 3 and Corollary 4 were read clause by clause on the page image. The proof of Theorem 3 was read and followed at the level of its steps; the computation behind Corollary 4 is not printed and was not checked. Nothing here is independently reviewed.

Proof pointer

Pages 302--303. With $n_r$ fixed for each $r$ and $\mathfrak S_m$ the product of the primes dividing some $n_r$, the form $F(m,1;z)$ lies modulo $m$ in the half-integral weight cusp space of level $576m\mathfrak S_m$, so Serre's theorem and the Shimura correspondence give a positive proportion of primes $\ell\equiv-1\pmod{576m\mathfrak S_m}$ with $F(m,1;z)\mid T(\ell^2)\equiv0\pmod m$. The formula (11) for $T(\ell^2)$, with quadratic reciprocity for the symbol $(n_r/\ell)$, then makes $p\big((mn_r\ell^2+1)/24\big)$ congruent modulo $m$ to $r$ times a sign independent of $r$ (display (13)), so for every large such $\ell$ these $m$ values meet every class. Counting such $\ell<X$, which are $\gg X/\log X$ in number, gives the $\sqrt X/\log X$ bound; the bound for $r=0$ comes from Theorem 1.

Dependencies

Theorems 1, 6 and 8 and Proposition 7 of the same paper; Serre's theorem ([S], 6.4) and the Shimura correspondence ([Sh], [Ci], [Ni]).

Bears on

No Erdős problem in this corpus. Newman's conjecture is not one of the problems recorded here.