arxiv: 2605.09807 · v1 · submitted 2026-05-10 · 🧮 math.NT

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On Ramanujan Primes for Hecke-Maass Cusp Forms

Shifan Zhao, Tinghao Huang

Pith reviewed 2026-05-12 02:46 UTC · model grok-4.3

classification 🧮 math.NT

keywords Ramanujan conjectureHecke-Maass cusp formsRamanujan primesHecke eigenvaluesnatural densitysimultaneous boundsanalytic estimates

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The pith

An explicit upper bound is given for the smallest prime where the Ramanujan conjecture holds simultaneously for two or three distinct Hecke-Maass cusp forms.

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

The Ramanujan conjecture for a primitive Hecke-Maass cusp form asserts that its Hecke eigenvalue λ_φ at any prime p satisfies |λ_φ(p)| ≤ 2. The paper establishes that for any two or three such distinct forms there exists an explicit upper bound on the smallest prime p at which this inequality holds for all the forms simultaneously. It further proves that for any finite collection of these forms the primes at which the conjecture is satisfied by at least one form have a positive lower natural density, with an explicit lower bound supplied for that density. The arguments rely on analytic control of the joint behavior of the Hecke eigenvalues across the forms.

Core claim

We determine an upper bound for the least prime p at which the Ramanujan conjecture holds for two or three distinct primitive Hecke-Maass cusp forms simultaneously. Moreover, given a set of distinct primitive Hecke-Maass cusp forms {ϕ_i}, we also provide a lower bound for the lower natural density of the set of primes at which the Ramanujan conjecture holds for at least one of the ϕ_i's.

What carries the argument

Simultaneous Ramanujan primes for a collection of Hecke-Maass cusp forms, defined as primes p where |λ_φ(p)| ≤ 2 holds for every form φ in the collection, with the bounds obtained via analytic estimates on the joint distribution of their Hecke eigenvalues.

Load-bearing premise

The analytic estimates or sieves used to control the Hecke eigenvalues of the Maass forms must hold with error terms strong enough to yield explicit bounds on the simultaneous primes.

What would settle it

Explicit computation of the smallest prime satisfying |λ_φ(p)| ≤ 2 simultaneously for two concrete Hecke-Maass cusp forms, if larger than the claimed upper bound, would disprove the result.

read the original abstract

For a primitive Hecke-Maass cusp form $\phi$ of level $N$ with the $n$-th Hecke eigenvalue $\lambda_{\phi}(n)$ and a prime number $p\nmid N$, the celebrated Ramanujan conjecture at $p$ asserts the following sharp upper bound: \[ |\lambda_{\phi}(p)| \leq 2. \] In this work, we determine an upper bound for the least prime $p$ at which the Ramanujan conjecture holds for two or three distinct primitive Hecke-Maass cusp forms simultaneously. Moreover, given a set of distinct primitive Hecke-Maass cusp forms $\{\phi_i\}$, we also provide a lower bound for the lower natural density of the set of primes at which the Ramanujan conjecture holds for at least one of the $\phi_i$'s.

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

This paper extends explicit bounds on the first prime satisfying the Ramanujan bound |λ(p)| ≤ 2 to Maass forms, with simultaneous results for two or three forms and a density lower bound.

read the letter

The main advance is an upper bound on the smallest prime p where |λ_φ(p)| ≤ 2 holds simultaneously for two or three distinct primitive Hecke-Maass cusp forms, plus a lower bound on the lower natural density of primes where the bound holds for at least one form in a finite set. This is a direct extension of earlier work on holomorphic forms to the Maass setting, and the abstract states the claims clearly without obvious circularity or fitted parameters. The approach appears to rely on standard analytic tools such as averages or sieves for Hecke eigenvalues, which is reasonable given the Kim–Sarnak bound already available for Maass forms. The results are presented as unconditional and effective, which is the right direction for this kind of concrete progress in analytic number theory. The soft spots are modest. The abstract gives no error terms, explicit constants, or method details, so it is impossible to judge how sharp the bounds are or how they scale with level and the number of forms. Any hidden dependence on unproven hypotheses would weaken the claims, though nothing in the stated results suggests that. The density bound is a natural companion but may not be strong enough to be useful outside narrow applications. The paper is aimed at specialists in automorphic forms and Hecke eigenvalues who need effective control over small primes. A reader already working on Ramanujan-type questions for Maass forms would find the simultaneous and density statements useful as a reference point. It deserves a serious referee because the extension is new in this context, the claims are falsifiable, and the underlying techniques are standard enough to check. I would send it to review rather than desk reject.

Referee Report

0 major / 3 minor

Summary. The manuscript claims to determine an explicit upper bound on the smallest prime p at which the Ramanujan conjecture |λ_φ(p)| ≤ 2 holds simultaneously for any two or three distinct primitive Hecke-Maass cusp forms. It further claims a lower bound on the lower natural density of the set of primes p for which the conjecture holds for at least one form in a given finite set of distinct primitive Hecke-Maass cusp forms.

Significance. If the stated bounds are unconditional and effective, the results would constitute a modest but concrete advance in the analytic theory of Hecke eigenvalues for Maass forms, extending single-form Ramanujan-type bounds to simultaneous and density settings. Explicit constants would allow direct comparison with known individual bounds such as Kim–Sarnak and could support numerical experiments on small-level forms.

minor comments (3)

[Abstract] The abstract and introduction should clarify whether the upper bounds on the least simultaneous prime are fully effective (i.e., numerically computable from the levels and spectral parameters of the forms) or merely existential; this distinction affects the utility of the result.
[Introduction] Notation for the lower natural density should be introduced explicitly (e.g., lim inf of the counting function) and the dependence of the density lower bound on the cardinality of the set {φ_i} should be stated clearly.
Any appeal to average estimates or sieve methods for Hecke eigenvalues should include a precise reference to the error term or zero-free region employed, even if standard.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the provided report, so we have no points to address individually at this stage. We remain available to incorporate any minor clarifications or adjustments once further details are supplied.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper claims explicit upper bounds on the smallest prime where the Ramanujan conjecture holds simultaneously for two or three distinct primitive Hecke-Maass cusp forms, plus a lower bound on the natural density of primes where it holds for at least one form in a finite set. These are presented as consequences of analytic techniques such as average estimates or sieves on Hecke eigenvalues. No equations, fitted parameters, self-definitions, or load-bearing self-citations appear in the abstract or described claims that reduce the results to tautologies or prior inputs by construction. The derivation chain relies on standard external tools in analytic number theory and is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on standard domain assumptions of the theory of automorphic forms and Hecke eigenvalues; no new free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

axioms (1)

domain assumption Standard analytic properties and eigenvalue bounds for primitive Hecke-Maass cusp forms of level N
Invoked implicitly when discussing λ_ϕ(p) and the Ramanujan conjecture at primes p not dividing N.

pith-pipeline@v0.9.0 · 5435 in / 1239 out tokens · 64880 ms · 2026-05-12T02:46:43.874514+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
We determine an upper bound for the least prime p at which the Ramanujan conjecture holds for two or three distinct primitive Hecke-Maass cusp forms simultaneously... lower bound for the lower natural density...

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