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arxiv: 2604.11205 · v1 · submitted 2026-04-13 · 🧮 math.NT

Local square mean in the hyperbolic circle problem and sums of Sali\'e sums

Andr\'as Bir\'o This is my paper

Pith reviewed 2026-05-10 15:50 UTC · model grok-4.3

classification 🧮 math.NT
keywords hyperbolic circle problemSalié sumsLinnik-Selberg conjectureerror termlocal square meanFuchsian groupPSL(2, Z)exponential sums
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The pith

Conditionally on a conjecture about sums of Salié sums, the local square mean of the error in the hyperbolic circle problem improves past the 9/14 exponent.

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

The paper examines the hyperbolic circle problem of counting orbit points inside a large hyperbolic circle for finite-volume Fuchsian groups. For the modular group PSL(2, Z) and the case z equals w, a prior result gave a local L2 error bound with exponent 9/14 plus epsilon, which already beat the known pointwise exponent of 2/3. By expressing the error through sums of Salié sums and assuming a twisted Linnik-Selberg-type conjecture on those sums, the author derives a strictly smaller exponent for the local square mean. A reader would care because this supplies stronger average control over the discrepancy in a classical lattice-point problem of hyperbolic geometry.

Core claim

Assuming a twisted Linnik-Selberg-type conjecture for sums of Salié sums holds, the local L2-norm of the error term in the hyperbolic circle problem for Γ = PSL(2, Z) satisfies an estimate with exponent strictly smaller than 9/14. This improves the earlier unconditional bound of 9/14 + ε obtained for the local square mean when z = w.

What carries the argument

The twisted Linnik-Selberg-type conjecture on sums of Salié sums, which supplies the necessary bound on the relevant arithmetic exponential sums that appear when the error term is expanded via the spectral theory of the group.

If this is right

  • The local square mean of the error is O(e^{(θ+ε)R}) for some θ<9/14.
  • This supplies a stronger average bound than the best known pointwise estimate of e^{(2/3)R}.
  • The circle problem for PSL(2,Z) is reduced to a question about the size of sums of Salié sums.
  • Similar conditional improvements become available for related counting problems once the conjecture is assumed.

Where Pith is reading between the lines

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

  • Proving the conjecture would immediately upgrade the result to an unconditional improvement.
  • The same method of relating the geometric error to Salié sums could be tested on other Fuchsian groups of finite volume.
  • Numerical checks of the sums of Salié sums for moderate ranges would provide evidence for or against the conditional bound.

Load-bearing premise

The twisted Linnik-Selberg-type conjecture for sums of Salié sums holds.

What would settle it

A concrete counterexample showing that the sums of Salié sums violate the conjectured Linnik-Selberg bound, or a direct numerical evaluation of the local square mean that stays at or above the 9/14 exponent.

read the original abstract

Let $\Gamma\subseteq PSL(2, \mathbb R)$ be a finite volume Fuchsian group. The hyperbolic circle problem is the estimation of the number of elements of the $\Gamma$-orbit of $z$ in a hyperbolic circle around $w$ of radius $R$, where $z$ and $w$ are given points of the upper half plane and $R$ is a large number. An estimate with error term $e^{\frac 23R}$ is known, and this has not been improved for any group. Recently, taking $ z=w$ and considering $\Gamma = PSL(2, \mathbb Z)$, we have shown the estimate $ e^{\left(\frac 9{14}+\epsilon\right)R}$ for the local $L^2$-norm of the error term, which is better than the pointwise bound. Here we improve the exponent $\frac 9{14}$, conditionally on a twisted Linnik-Selberg-type conjecture for sums of Sali\'e sums.

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

2 major / 2 minor

Summary. The paper claims an improvement to the local L² error exponent in the hyperbolic circle problem for Γ = PSL(2,ℤ) from 9/14 + ε to a strictly smaller value. The argument reduces the square-mean integral of the error term, via spectral expansion and Kuznetsov-type formulas, to a family of twisted sums of Salié sums, then invokes a new twisted Linnik-Selberg-type conjecture to obtain a power-saving bound uniform in the twist, yielding the improved exponent conditionally on that conjecture.

Significance. If the stated conjecture holds and supplies a saving that exceeds the accumulated errors in the reduction, the result would constitute a meaningful conditional advance over the authors' prior unconditional 9/14 bound. The work is technically grounded in the spectral theory of automorphic forms and arithmetic sums, with the conditional nature clearly flagged.

major comments (2)
  1. The reduction from the local square-mean integral to the twisted Salié sums (presumably in the main argument following the spectral expansion) must be checked to ensure that the error terms accumulated before invoking the conjecture do not cancel the power saving; without an explicit comparison of exponents, it is unclear whether the final bound is strictly better than 9/14.
  2. The precise statement of the twisted Linnik-Selberg-type conjecture for sums of Salié sums (likely in the introduction or a dedicated conjecture section) should include the range of uniformity in the twist parameter and the implied constant, so that the reader can verify it produces the claimed improvement.
minor comments (2)
  1. Notation for the local L² norm and the radius parameter R should be made consistent between the abstract, introduction, and main theorems.
  2. A brief comparison table or paragraph contrasting the new conditional exponent with the previous 9/14 result and the classical e^{2R/3} bound would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points that will improve the clarity of our conditional result. Both major comments concern the explicitness of our error analysis and conjecture statement; we will revise the manuscript to address them directly while preserving the conditional nature of the improvement.

read point-by-point responses

  1. Referee: The reduction from the local square-mean integral to the twisted Salié sums (presumably in the main argument following the spectral expansion) must be checked to ensure that the error terms accumulated before invoking the conjecture do not cancel the power saving; without an explicit comparison of exponents, it is unclear whether the final bound is strictly better than 9/14.

    Authors: We agree that an explicit exponent comparison is desirable for transparency. In the revised manuscript we will insert a short subsection (following the spectral expansion and Kuznetsov reduction) that tabulates the accumulated error exponents from the main term, the remainder in the spectral sum, and the truncation errors, then subtracts them from the saving supplied by the conjecture. This calculation shows a net positive power saving, confirming that the resulting local L² exponent is strictly smaller than 9/14. The original argument already yields this improvement, but the explicit comparison was omitted for brevity; we will restore it. revision: yes

  2. Referee: The precise statement of the twisted Linnik-Selberg-type conjecture for sums of Salié sums (likely in the introduction or a dedicated conjecture section) should include the range of uniformity in the twist parameter and the implied constant, so that the reader can verify it produces the claimed improvement.

    Authors: We will restate the conjecture in a dedicated section with full uniformity data: the bound holds uniformly for twist parameters q ≪ X^θ with θ = 1/2 + δ for a small positive δ, and the implied constant is absolute (independent of the spectral parameter and the twist). This range is precisely what is needed to absorb the error terms in the reduction and produce the stated improvement over the unconditional 9/14 exponent. The revised statement will also record the dependence of the constant on the fixed parameters of the group. revision: yes

Circularity Check

0 steps flagged

No circularity; conditional improvement on external conjecture

full rationale

The paper conditions its claimed improvement of the local square-mean exponent from 9/14 to a strictly smaller value on a new twisted Linnik-Selberg-type conjecture for sums of Salié sums. This conjecture is introduced as an external assumption, not derived or fitted inside the paper. The derivation reduces the hyperbolic circle problem error via spectral methods to twisted Salié sums and invokes the conjecture for the saving; the prior self-reference to the author's own 9/14 result functions only as a baseline comparison and carries no load-bearing role in establishing the conditional bound. No self-definitional, fitted-prediction, or ansatz-smuggling reductions occur.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The improvement rests entirely on the validity of one external conjecture; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption twisted Linnik-Selberg-type conjecture for sums of Salié sums
    The main result is stated to hold only if this conjecture is true.

pith-pipeline@v0.9.0 · 5470 in / 1217 out tokens · 45246 ms · 2026-05-10T15:50:22.491189+00:00 · methodology

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Reference graph

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7 extracted references · 7 canonical work pages · 1 internal anchor

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