[Paper Review] Distribution of the partition function modulo m

Ken Ono|ArXiv.org|Aug 17, 2000

Advanced Mathematical Identities16 references18 citations

TL;DR

This paper establishes that for every prime m ≥ 5, there exists a positive proportion of primes ℓ such that the partition function p(n) satisfies p(mℓ³n + 1/24) ≡ 0 (mod m) for all n coprime to ℓ. Using modular forms and the Shimura correspondence, it proves Erdős’ conjecture on the infinitude of primes dividing some p(n), and confirms Newman’s conjecture for all primes m < 1000 except possibly m = 3, via the discovery of 'Ramanujan cycles'—eventually periodic generating functions modulo m.

ABSTRACT

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).

Motivation & Objective

• To resolve the long-standing debate on whether there are finitely or infinitely many Ramanujan-type congruences p(an + b) ≡ 0 (mod m).
• To prove Erdős’ conjecture that for every prime m, there exists at least one n with p(n) ≡ 0 (mod m).
• To verify Newman’s conjecture that for every prime m, every residue class mod m is hit infinitely often by p(n) mod m.
• To establish the existence of 'Ramanujan cycles'—eventually periodic generating functions modulo m for the partition function.
• To provide a uniform, systematic framework using modular forms to analyze the distribution of p(n) modulo m.

Proposed method

• Leverages the theory of half-integral weight modular forms and their reductions modulo prime m, showing that generating functions F(m,k;z) lie in finite-dimensional spaces over F_m.
• Applies the Shimura correspondence to relate the partition function modulo m to cusp forms of weight (m²−m−1)/2 with Nebentypus character.
• Uses Sturm’s theorem to verify congruences between modular forms via finite computations on Fourier coefficients.
• Employs the U-operator and T-operator actions on modular forms to analyze periodicity in the sequence F(m,k;z) as k increases.
• Introduces the concept of 'good primes' m for which certain initial conditions ensure full coverage of residue classes mod m.
• Uses computational tools developed by Haglund and Haynal to verify that all primes m < 1000 (except possibly m = 3) are good, thereby confirming Newman’s conjecture.

Experimental results

Research questions

• RQ1 Are there infinitely many primes m for which there exists at least one n with p(n) ≡ 0 (mod m)?
• RQ2 For a given prime m, does every residue class r (mod m) occur infinitely often in the sequence p(n) mod m?
• RQ3 Can the generating functions F(m,k;z) = ∑ p(mᵏn + 1/24) qⁿ mod m exhibit periodic behavior in k?
• RQ4 What is the structure of the space of modular forms modulo m that govern the partition function?
• RQ5 Can the existence of Ramanujan-type congruences be systematically explained via modular forms and Hecke operators?

Key findings

• For every prime m ≥ 5, there is a positive proportion of primes ℓ such that p(mℓ³n + 1/24) ≡ 0 (mod m) for all n coprime to ℓ.
• Erdős’ conjecture is confirmed: for every prime m, there exists at least one n with p(n) ≡ 0 (mod m), and the number of such n ≤ X is ≫ X for m ≥ 5.
• Newman’s conjecture is true for all primes m < 1000 except possibly m = 3, with the number of solutions n ≤ X for each residue r mod m being ≫ √X / log X (for 1 ≤ r ≤ m−1) and ≫ X (for r = 0).
• The generating functions F(m,k;z) are eventually periodic in k modulo m, forming 'Ramanujan cycles'—a key structural discovery.
• Explicit congruences are derived for m = 13, 17, 19, 23, such as p(594·13ⁿ + 111247) ≡ 0 (mod 13), and p(23²ᵏ⁺¹(24n+1)+1)/24 ≡ 5ᵏg(n) (mod 23).
• For each prime m ≥ 5, there exist integers N(m) and P(m) ≤ 48(m³−2m−1) such that p(mⁱn + 1/24) ≡ p(mⁱ⁺ᴾ⁽ᵐ⁾n + 1/24) (mod m) for all i > N(m) and all n ≥ 0.