Distribution of the partition function modulo $m$

By Ken Ichi Ono. 2000

240 citations recorded by the source corpus.

Ramanujan (and others) proved that the partition function satisfies a number of striking congruences modulo powers of 5, 7 and 11. A number of further congruences were shown by the works of Atkin, O'Brien, and Newman. In this paper we prove that there are infinitely many such congruences for every prime modulus exceeding 3. In addition, we provide a simple criterion guaranteeing the truth of Newman's conjecture for any prime modulus exceeding 3 (recall that Newman's conjecture asserts that the partition function hits every residue class modulo a given integer M infinitely often).

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/f4e69d8f7b24256c973d20869cd3787142e717c3

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

Citations in the corpus

Dyson’s ranks and Maass forms (2010, methodology): “5 now follows mutatis mutandis as in the proof of Theorem 1 of [24] (see the top of page 301 of [24]).”
Congruences for Andrews' Smallest Parts Partition Function and New Congruences for Dyson's Rank (2007, unknown): “All these results generalize the analogue for p(n) proved by Ono [ 26 ].”
DYSON'S RANK, OVERPARTITIONS, AND WEAK MAASS FORMS (2007, methodology): “hat p(0) := 1, and p(n) := 0 for n∈ N 0). Ramanujan proved that for every positive integer n, we have: p(5n+4) ≡ 0 (mod 5), p(7n+5) ≡ 0 (mod 7), p(11n+6) ≡ 0 (mod 11). (1.1) In a celebrated paper Ono [27] treated these kinds of congruences systematically (also see [29]). Combining Shimura’s theory o”
RANK AND CONGRUENCES FOR OVERPARTITION PAIRS (2008, background): “Using theta functions and q-series identities, Atkin [6] and Watson [38] later proved generalisations for powers of 5, 7, and 11. In a celebrated paper Ono [ 32 ] treated these kinds of congruences systematically.”
ℓ-Divisibility of ℓ-regular partition functions (2009, background): “for every nonnegative integer n. Recently K. Ono [ 4 ] showed that there exist congruences for p(n) modulo every prime greater than 3, and S. Ahlgren and M. Boylan [1] demonstrated the uniqueness of the Ramanujan congruences among these by proving that if � is prime and there exists an integer a suc”
Addition and counting: the arithmetic of partitions (2001, background): “The 1960s, however, witnessed tantilizing discoveries of further examples (notably by Atkin, Newman, and O ’Brien).”
Distribution of a certain partition function modulo powers of primes (2011, methodology): “Note that δm is related to the frequently used symbol δm defined in some earlier papers such as [10, 11]: δm := m − 1 24 = δm 3 .”
ON THE EXPLICIT CONSTRUCTION OF HIGHER DEFORMATIONS OF PARTITION STATISTICS (2007, background): “[12] K. Bringmann and K. Ono, The f(q) mock theta function conjecture and partition ranks, Invent.”
Arithmetic properties of the partition function (2003, background): “In this section we recall some facts about modular forms modulo � . O ne may consult, for example, [SwD] or [S] for details.”
Unearthing the visions of a master: harmonic Maass forms and number theory (2008, methodology): “1 of [201]), generalizing earlier observations of Ahlgren and the author [3, 162], implies that there is a sufficiently large integer m for which ∑”
Partitions: At the Interface of q-Series and Modular Forms (2003, background): “Seminal papers in this vein include the Hardy-Ramanujan paper on p(n) [20], the proofs of the Ramanujan congruences by Watson [41] and Atkin [7], and the many recent contributions by Ono [28, 29 ] to name only a few.”
ARITHMETIC OF ℓ-REGULAR PARTITION FUNCTIONS (2008, background): “Recently Ono [7] made a remarkable breakthrough by showing that congruences for p(n) are widespread, proving that for every prime m ≥ 5, there exist infinitely many non-nested arithmetic progressions An + B such that for every nonnegative integer n, p(An+B) ≡ 0 (mod m).”
Ramanujan’s Unpublished Manuscript on the Partition and Tau Functions with Proofs and Commentary (2001, methodology): “On the other hand, the second author [73] has shown that if M ≥ 5 is prime, then #{n ≤ X : p(n) ≡ 0 (mod M)} ÀM X.”
Congruences for the coefficients of weakly holomorphic modular forms (2006, methodology): “Recently, Ahlgren and Ono (in [1, 2, 12]) showed that if M is any positive integer coprime to 6, then there exist infinitely many congruences of the form p(An + B) ≡ 0 (mod M).”
Arithmetic of the 13-regular partition function modulo 3 (2011, result): “Recent works of Ahlgren and Ono, [1, 3, 9 ], show that for all moduli m coprime to 6, there are infinitely many arithmetic progressions An + B such that”