The Second Moment of Rankin-Selberg L-Functions in Conductor-Dropping Regimes
Pith reviewed 2026-05-23 05:40 UTC · model grok-4.3
The pith
An asymptotic formula holds for the second moment of Rankin-Selberg L-functions from pairs of equal-weight holomorphic Hecke cusp forms in conductor-dropping regimes.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We prove an asymptotic formula for the second moment of L-functions associated to the Rankin-Selberg convolution of two holomorphic Hecke cusp forms with equal weight.
What carries the argument
The Rankin-Selberg L-function attached to a pair of equal-weight holomorphic Hecke cusp forms, whose second moment is given by an asymptotic formula in conductor-dropping regimes.
Load-bearing premise
The asymptotic applies only to holomorphic Hecke cusp forms of equal weight in conductor-dropping regimes.
What would settle it
A calculation of the second moment for a concrete pair of equal-weight holomorphic cusp forms in a conductor-dropping regime that deviates from the predicted main term by more than the claimed error would falsify the result.
read the original abstract
We prove an asymptotic formula for the second moment of $L$-functions associated to the Rankin-Selberg convolution of two holomorphic Hecke cusp forms with equal weight.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proves an asymptotic formula for the second moment of L-functions associated to the Rankin-Selberg convolution of two holomorphic Hecke cusp forms with equal weight, specifically in conductor-dropping regimes.
Significance. If the result holds, it supplies a new asymptotic in a specialized regime for these moments, which may have applications to subconvexity bounds or value-distribution questions for automorphic L-functions. The self-contained analytic proof and explicit restriction to equal weights and the named regime are strengths.
minor comments (2)
- The abstract is concise but does not indicate the shape of the main term or error term in the asymptotic; adding one sentence on this point would improve accessibility without lengthening the paper unduly.
- Notation for the forms (e.g., f and g) and the precise definition of the conductor-dropping regime should be introduced at the first use in the introduction for readers who skip directly to the statement of the main theorem.
Simulated Author's Rebuttal
We thank the referee for their positive assessment of the manuscript and for recommending minor revision. No specific major comments appear in the report, so we have no point-by-point responses to provide. We are pleased that the referee notes the result supplies a new asymptotic in the conductor-dropping regime and recognizes the strengths of the self-contained analytic proof.
Circularity Check
No significant circularity detected
full rationale
The paper presents a self-contained analytic proof of an asymptotic formula for the second moment of Rankin-Selberg L-functions attached to holomorphic Hecke cusp forms of equal weight in conductor-dropping regimes. The central claim is a derivation from standard methods in analytic number theory, with the equal-weight and regime restrictions explicitly stated as part of the result rather than hidden assumptions. No load-bearing steps reduce by construction to fitted inputs, self-citations, or ansatzes; the argument is internally consistent and externally verifiable without circular reduction.
discussion (0)
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