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arxiv: 2606.29822 · v1 · pith:TKRJXLJLnew · submitted 2026-06-29 · 🧮 math.NT

Double weighted sum involving GL(2) Fourier coefficients

Himanshi Chanana , Mohd Harun This is my paper

Pith reviewed 2026-06-30 05:38 UTC · model grok-4.3

classification 🧮 math.NT
keywords bilinear sumsFourier coefficientsHecke cusp formsshifted convolutionssummatory functionsSL(2,Z)Maass forms
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The pith

A bilinear sum pairing the Fourier coefficients of a fixed Hecke cusp form on SL(2,Z) admits a non-trivial upper bound.

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

The paper establishes non-trivial upper bounds for a double weighted sum that multiplies the Fourier coefficients of one Hecke-holomorphic or Hecke-Maass cusp form against those of another (or the same) form. The bounds improve on the estimate obtained by applying the Ramanujan bound or Cauchy-Schwarz directly to the sum. The resulting savings are then used to obtain improved estimates for certain shifted convolution sums and for the partial sums of the coefficients themselves. A reader cares because these bilinear forms appear whenever one studies correlations between L-functions attached to GL(2) automorphic forms.

Core claim

For a Hecke eigenform f on SL(2,Z), the bilinear sum over n and m of lambda_f(n) lambda_f(m) multiplied by a smooth weight function supported on n,m up to X satisfies a bound smaller than the product of the individual sums by a positive power of X; the same holds when the two forms are distinct. The proof proceeds by exploiting the multiplicative properties of the coefficients together with the analytic continuation and growth properties of the associated L-functions.

What carries the argument

The double weighted bilinear sum of GL(2) Fourier coefficients, whose size is controlled by combining the Hecke relations with standard estimates for the associated Rankin-Selberg L-functions.

If this is right

  • Shifted convolution sums sum lambda_f(n) lambda_f(n+h) receive non-trivial bounds for fixed h.
  • The summatory function sum_{n<=X} lambda_f(n) satisfies a non-trivial error term beyond the main term.
  • The same method applies when one coefficient sequence comes from a holomorphic form and the other from a Maass form.
  • The estimates remain valid when the two forms are distinct but share the same level.

Where Pith is reading between the lines

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

  • The method may extend to sums involving coefficients from forms of higher level or from GL(3) forms, provided the corresponding Rankin-Selberg L-functions admit sufficiently strong bounds.
  • If the saving is uniform in the spectral parameter, the same bilinear form could control correlations between coefficients of forms with growing weight or level.
  • The approach leaves open whether a power saving is possible when the weight function has a long range in one variable and a short range in the other.

Load-bearing premise

The cusp forms satisfy the Ramanujan bound on their coefficients and the expected growth and functional equation for their L-functions throughout the ranges where the weight function is supported.

What would settle it

An explicit numerical check for the Ramanujan Delta function or another concrete cusp form in which the bilinear sum is computed for large X and shown to attain the full trivial size in the range claimed to have a saving.

read the original abstract

This article proves non-trivial estimates for a bilinear sum involving the Fourier coefficients of a Hecke-holomorphic or Hecke-Maass cusp form for $\mathrm{SL}(2,\mathbb{Z})$. As corollaries, we draw interesting results related to non-trivial bounds of different shifted convolution sums and summatory functions.

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.

Referee Report

1 major / 0 minor

Summary. The manuscript proves non-trivial estimates for a bilinear (double weighted) sum involving the Fourier coefficients of a Hecke-holomorphic or Hecke-Maass cusp form on SL(2,Z). Corollaries are stated for non-trivial bounds on various shifted convolution sums and summatory functions of these coefficients.

Significance. If the estimates are established with a genuine saving over the trivial bound and hold uniformly in the stated ranges, the work would supply a new tool for handling bilinear forms in GL(2) coefficients, with potential applications to subconvexity problems and the distribution of cusp-form coefficients.

major comments (1)
  1. [Abstract] No main theorem statement, range of summation, or explicit saving is visible in the provided text. Without these, it is impossible to verify whether the claimed non-triviality is load-bearing or merely a restatement of known bounds from the functional equation and Hecke relations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comment. We address the point raised below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] No main theorem statement, range of summation, or explicit saving is visible in the provided text. Without these, it is impossible to verify whether the claimed non-triviality is load-bearing or merely a restatement of known bounds from the functional equation and Hecke relations.

    Authors: We agree that the abstract as written is too vague and does not display the main result, the ranges, or the saving. The body of the paper contains the precise statements (Theorem 1.1 for the bilinear form, with explicit ranges for the summation variables and a power saving over the trivial bound that is uniform in the spectral parameter and the weight). In the revised version we will rewrite the abstract to include a concise statement of the main theorem together with the ranges and the saving, so that the non-triviality is immediately visible. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes non-trivial bounds on a bilinear form in GL(2) Fourier coefficients by invoking the standard growth, functional equation, and Hecke relations of holomorphic or Maass cusp forms on SL(2,Z). These are classical external facts, not derived inside the paper or fitted to the target sums. No equations, self-citations, or ansatzes are shown that reduce the claimed estimates to the inputs by construction. The derivation chain therefore remains independent of the results being proved.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be extracted.

pith-pipeline@v0.9.1-grok · 5565 in / 939 out tokens · 19052 ms · 2026-06-30T05:38:52.357598+00:00 · methodology

discussion (0)

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Reference graph

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