Ramanujan Congruences for Fractional Partition Functions

Erin Bevilacqua; Kapil Chandran; Yunseo Choi

arxiv: 1907.06716 · v1 · pith:INJ4W53Nnew · submitted 2019-07-15 · 🧮 math.NT · math.CO

Ramanujan Congruences for Fractional Partition Functions

Erin Bevilacqua , Kapil Chandran , Yunseo Choi This is my paper

Pith reviewed 2026-05-24 21:13 UTC · model grok-4.3

classification 🧮 math.NT math.CO

keywords fractional partition functionsRamanujan congruencesnon-ordinary primesDedekind eta-functionpartition generating functionscongruences modulo integersrational exponents

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The pith

Non-ordinary primes characterize congruences for fractional partition functions modulo any integer.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a general framework for congruences of the fractional partition functions p_α(n) when α is rational, using the generating function (q;q)_∞^α. Ramanujan found such congruences only for the ordinary partition function and the primes 5, 7, and 11; the authors extend this reach by applying the theory of non-ordinary primes together with Dedekind eta-function identities. The framework is designed to work for congruences modulo arbitrary integers rather than selected primes. A sympathetic reader would care because the approach supplies both a characterization and explicit new families, including the stated result for modulus 17 squared.

Core claim

The authors show that the theory of non-ordinary primes applies to the generating functions (q;q)_∞^α for rational α via Dedekind eta-function identities, producing a general framework that characterizes congruences modulo any integer and thereby proves new results such as p_{57/61}(17²n-3)≡0 mod 17².

What carries the argument

The theory of non-ordinary primes applied to the generating function (q;q)_∞^α through Dedekind eta-function identities

If this is right

Congruences for fractional partition functions exist for every modulus, not merely the primes 5, 7, and 11.
Infinite families of such congruences can be constructed for many choices of rational α.
The same technique yields explicit results such as the congruence modulo 17² given in the paper.
The framework supplies a systematic way to decide whether a given arithmetic progression carries a congruence for a chosen α and modulus.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same non-ordinary-prime approach might be tested on other q-series whose coefficients arise from powers of the Euler function.
Direct computation of p_{57/61}(k) for the first few dozen values of k in the arithmetic progression could provide an independent check of the claimed congruence.
If the framework is robust, it could be applied to generating functions attached to other eta-quotients or to mock theta functions.

Load-bearing premise

The theory of non-ordinary primes applies directly to the generating functions (q;q)_∞^α for rational α via Dedekind eta-function identities to yield the stated congruences for arbitrary moduli.

What would settle it

An explicit integer n for which p_{57/61}(17²n-3) is not divisible by 17², or a rational α and modulus m for which the non-ordinary-prime framework produces no congruences at all.

read the original abstract

For rational $\alpha$, the fractional partition functions $p_\alpha(n)$ are given by the coefficients of the generating function $(q;q)^\alpha_\infty$. When $\alpha=-1$, one obtains the usual partition function. Congruences of the form $p(\ell n + c)\equiv 0 \pmod{\ell}$ for a prime $\ell$ and integer $c$ were studied by Ramanujan. Such congruences exist only for $\ell\in\{5,7,11\}.$ Chan and Wang [4] recently studied congruences for the fractional partition functions and gave several infinite families of congruences using identities of the Dedekind eta-function. Following their work, we use the theory of non-ordinary primes to find a general framework that characterizes congruences modulo any integer. This allows us to prove new congruences such as $p_\frac{57}{61}(17^2n-3)\equiv 0 \pmod{17^2}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims a general framework for congruences on fractional partition functions p_alpha(n) via non-ordinary primes after eta rewrites, with one new explicit example, but the modular-form reduction for fractional weights is the part that needs checking.

read the letter

The main takeaway is that this work extends Ramanujan congruences to p_alpha(n) for rational alpha by rewriting the generating function (q;q)_infty^alpha as an eta-product and then using non-ordinary primes to get congruences modulo any integer, not just 5, 7, 11. They give a concrete new one: p_{57/61}(17^2 n - 3) ≡ 0 mod 17^2. That is the actual advance over Chan-Wang, who stopped at infinite families for specific cases. The paper does a clean job of identifying the right tool (non-ordinary primes) and stating the target result explicitly. The citation pattern looks standard and the circularity burden is low since it rests on existing eta identities and the external theory of non-ordinary primes. The soft spot is exactly the load-bearing step the stress-test flagged. Non-ordinary primes are defined for Hecke eigenvalues of forms with integer or half-integral weight; here the naive weight for alpha = 57/61 is 57/122. The paper must show that the eta identities produce a form of appropriate level and weight where the non-ordinary condition directly controls the coefficients modulo arbitrary m, including 17^2. The abstract gives no derivation or explicit eta-product, so it is impossible to see whether the reduction works without gaps or extra conditions. If the full proofs close that gap cleanly, the framework holds; if they do not, the claim to characterize congruences for any modulus weakens to something more case-by-case. This is narrow-scope number theory, aimed at people already working on partition congruences and eta-products. A specialist reader will get a usable general method if the details check out. It is worth sending to a serious referee so the technical application of non-ordinary primes to these fractional cases can be verified.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a general framework for Ramanujan-style congruences of fractional partition functions p_α(n), defined via coefficients of (q;q)_∞^α for rational α, by applying the theory of non-ordinary primes to generating functions rewritten via Dedekind eta-function identities. It claims this characterizes congruences modulo arbitrary integers and proves new examples, including p_{57/61}(17²n-3)≡0 mod 17².

Significance. If the central reduction to forms where non-ordinary primes control coefficients holds, the work would meaningfully extend classical partition congruence theory and the recent eta-identity approach of Chan-Wang to arbitrary moduli and fractional α, offering a systematic method rather than case-by-case identities.

major comments (2)

[Introduction / framework section] The load-bearing step is the claim that non-ordinary prime theory (typically for integer- or half-integral-weight newforms) applies after rewriting (q;q)_∞^α as an eta-product for rational α. For α=57/61 the naive weight is 57/122; the manuscript must explicitly exhibit the eta-product identity, the resulting weight, level, and character, and verify that the non-ordinary condition at 17 implies the stated coefficient congruence modulo 17². No such derivation appears in the abstract or the described framework.
[The General Framework] The general-framework claim (characterizing congruences modulo any integer) rests on the same reduction working uniformly for arbitrary m. The single explicit example does not substitute for a proof that the eta-identity step produces a form whose Hecke eigenvalues are controlled by non-ordinary primes for every prime power modulus.

minor comments (2)

[Introduction] Notation for the fractional partition function p_α(n) and the generating function (q;q)_∞^α should be introduced with a displayed equation early in the introduction.
[Introduction] The reference to Chan and Wang [4] should include a brief statement of which eta identities are being extended.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and for identifying the key points that require clarification. We address each major comment below. Where the manuscript is missing explicit derivations, we will incorporate them in revision.

read point-by-point responses

Referee: [Introduction / framework section] The load-bearing step is the claim that non-ordinary prime theory (typically for integer- or half-integral-weight newforms) applies after rewriting (q;q)_∞^α as an eta-product for rational α. For α=57/61 the naive weight is 57/122; the manuscript must explicitly exhibit the eta-product identity, the resulting weight, level, and character, and verify that the non-ordinary condition at 17 implies the stated coefficient congruence modulo 17². No such derivation appears in the abstract or the described framework.

Authors: We agree that the explicit eta-product identity, weight, level, and character for α=57/61 must be displayed, together with the verification that the non-ordinary prime at 17 produces the claimed congruence modulo 17². The current manuscript states the framework but omits this concrete derivation. In the revised version we will insert the required identity (obtained via the methods of Chan-Wang) and the subsequent modular-form data, confirming applicability of the non-ordinary prime criterion even though the naive weight 57/122 is fractional. revision: yes
Referee: [The General Framework] The general-framework claim (characterizing congruences modulo any integer) rests on the same reduction working uniformly for arbitrary m. The single explicit example does not substitute for a proof that the eta-identity step produces a form whose Hecke eigenvalues are controlled by non-ordinary primes for every prime power modulus.

Authors: The framework asserts that, for any rational α, an eta-product representation exists whose associated modular form (possibly of non-integral weight) has Hecke eigenvalues governed by non-ordinary primes at primes dividing the modulus. While the paper illustrates the method with one new congruence, it does not contain a uniform proof that the reduction step succeeds for every prime power. We will therefore expand the general-framework section with an outline of the uniform reduction argument and, if space permits, a second small example for a different prime power. revision: partial

Circularity Check

0 steps flagged

No significant circularity; framework applies external non-ordinary prime theory via eta identities

full rationale

The derivation chain starts from the generating function (q;q)_∞^α for rational α, rewrites it using known Dedekind eta-function identities (citing Chan-Wang), and invokes the independent theory of non-ordinary primes on the resulting modular forms. No step defines a quantity in terms of itself, renames a fitted parameter as a prediction, or reduces the central claim to a self-citation chain. The new congruences (e.g., for α=57/61 mod 17²) are applications of the external theory rather than tautological restatements of the inputs. The paper is self-contained against external benchmarks from modular forms.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard properties of Dedekind eta-function and applicability of non-ordinary prime theory to fractional generating functions; no free parameters or invented entities apparent from abstract.

axioms (2)

standard math Identities of the Dedekind eta-function hold and connect the generating function (q;q)_∞^α to modular forms.
Invoked following Chan and Wang to study congruences.
domain assumption Theory of non-ordinary primes characterizes congruences for these generating functions modulo any integer.
Central to the general framework described.

pith-pipeline@v0.9.0 · 5707 in / 1259 out tokens · 21342 ms · 2026-05-24T21:13:53.111565+00:00 · methodology

Ramanujan Congruences for Fractional Partition Functions

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)