A proof of Subbarao's conjecture

Abstract

Let pðnÞ denote the ordinary partition function. Subbarao conjectured that in every arithmetic progression r ðmod tÞ there are infinitely many integers N 1 r ðmod tÞ for which pðNÞ is even, and infinitely many integers M 1 r ðmod tÞ for which pðMÞ is odd. In the even case the conjecture was settled by Ken Ono. In this paper we prove the odd part of the conjecture which together with Ono's result implies the full conjecture. We also prove that for every arithmetic progression r ðmod tÞ there are infinitely many integers N 1 r ðmod tÞ such that pðNÞ E 0 ðmod 3Þ, which settles an open problem posed

Linear incongruences for generalized eta-quotients

“…See also [3] and the sources contained therein for further results. However, recent work of Radu [9] proved that there are no linear congruences of p(n) modulo 2 and 3, affirming a famous conjecture of Subbarao. The main ingredients of his proof are skillful computations and the q-expansion principle due to Deligne and Rapoport [4].…”

Mock theta functions and weakly holomorphic modular forms modulo 2 and 3

“…(3.25) By the argument in [28,Theorem 4.2], we see that as a ranges over integers with (a, 6m) = 1, the quantity…”

“…We require a fact proved by Deligne and Rapoport (see [14,VII,Corollary 3.12] or [28,Corollary 5.3]): If f ∈ M k (Γ 1 (N)) has coefficients in Z[ζ N ] then the same is true of f | k γ for each γ ∈ Γ 0 (N).…”

“…The current results are far from this expectation; for example the best result for the number of odd values of p(n) is due to Nicolas [23], who obtained the bound √ x(log log x) K / log x for any K. An old conjecture of Subbarao [29] states that there are no linear congruences for p(n) modulo 2. In striking recent work, Radu [28] has proved Subbarao's conjecture, as well as its analog modulo 3, with clever and technical arguments. In this paper we will adapt the methods of Radu's paper to prove non-vanishing results for the coefficients of mock theta functions and weakly holomorphic modular forms of certain types.…”

A mod ℓ Atkin–Lehner theorem and applications

“…Despite the difficulty of this conjecture, there are some results. For example, Subbarao's [12] Conjecture has been proved by the works of the first author and Radu [8,11]. For every progression r (mod t), there are infinitely many m (resp.…”