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arxiv: 2604.17077 · v1 · submitted 2026-04-18 · 🧮 math.NT

The Limiting Distribution of Elliptic Dedekind Sums

Matteo Bordignon , Paolo Minelli This is my paper

Pith reviewed 2026-05-10 06:23 UTC · model grok-4.3

classification 🧮 math.NT
keywords elliptic Dedekind sumsGaussian limiting distributionDedekind sumsIto conjecturecomplex latticeslimiting distributionsnumber theory
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The pith

Suitably normalized elliptic Dedekind sums have a Gaussian limiting distribution.

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

The paper examines elliptic Dedekind sums as generalizations of classical ones to complex lattices. It proves that with suitable normalization these sums have a Gaussian limiting distribution. This is then used to prove Ito's conjecture. A sympathetic reader cares because such limits describe the average behavior of number theoretic functions over families of lattices, enabling asymptotic predictions.

Core claim

We consider elliptic Dedekind sums that were introduced by Sczech as generalizations of the classical ones to complex lattices. We prove that these sums -- suitably normalized -- have a Gaussian limiting distribution. As an application, we prove a conjecture due to Ito.

What carries the argument

Suitably normalized elliptic Dedekind sums, where the normalization is chosen to yield the Gaussian limit through analytic or probabilistic machinery.

Load-bearing premise

The specific normalization chosen for the elliptic Dedekind sums is the one that produces the Gaussian limit, and the sums are defined in a manner that permits application of the required probabilistic or analytic machinery.

What would settle it

Sampling many normalized sums over lattices of increasing size and finding that their empirical distribution deviates from Gaussian, for example by showing skewness or kurtosis inconsistent with normality, would disprove the claim.

Figures

Figures reproduced from arXiv: 2604.17077 by Matteo Bordignon, Paolo Minelli.

Figure 1
Figure 1. Figure 1: Plot of D˜ (a, c)/ √ log X log log X + o(1) over the set K•,D(X) for X = 2500 and D = 2. 1.2. Overview. The proof of Vardi’s result Theorem 1.1 rests crucially on a link between Dedekind sums and Kloosterman sums. Thus, the spectral theory of automorphic forms comes into play in handling certain sums of Kloosterman sums (see also [23]). On the other hand, relatively recently, Theorem 1.1 was reproved, as t… view at source ↗
Figure 2
Figure 2. Figure 2: The fundamental domains ID for D ∈ {2, 7, 11}, with algebraic integers marked in red (origin removed). Remark 2.3. Plainly, the sets Oα above play the same role for the system (ID, G) as the intervals ( 1 n+1 , 1 n ] do for ((0, 1], T ). However, there are some important differences. For the Gauss map we have T (( 1 n+1 , 1 n ]) = (0, 1] for all n ≥ 1. On the other hand, as remarked in [31], there are α su… view at source ↗
Figure 3
Figure 3. Figure 3: The partition P for D = 2, 7, 11. The parts in P[2] are exactly the open subsets bounded by red arcs or purple lines. The plot was realized using the sets of lines and circles explicitly given in [20] In view of the above partition, the operator we are going to define will require, for computations, a certain understanding of the restrictions of the inverse branches to the various parts of P. In what follo… view at source ↗
Figure 4
Figure 4. Figure 4: Labeling of the elements (Pi)1≤i≤8 in the D = 2 case, as in Lemma 8.2. Lemma 8.2 (The rectangular case: D = 2). There is an n0 ≥ 1 and a list P1≤i≤8 of elements in P[2] such that for max(|r|, |n|) ≥ n0, the interior of the sets Vr,n is contained in exactly one of these two-dimensional parts. If these parts are labeled as in [PITH_FULL_IMAGE:figures/full_fig_p029_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Central subdivisions into relevant parts (Pi)1≤i≤6, with auxiliary lines depicted in blue. Remark 8.4. As we will show later, the contribution of the parts Vr,n with rn = 0 is negligible, thus only the parts (Pi)1≤i≤4 are relevant in the case D = 2, and all of them have the same µ-measure. This is not the case for D = 7 or D = 11. Indeed, as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p030_5.png] view at source ↗
read the original abstract

We consider elliptic Dedekind sums that were introduced by Sczech as generalizations of the classical ones to complex lattices. We prove that these sums -- suitably normalized -- have a Gaussian limiting distribution. As an application, we prove a conjecture due to Ito.

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 manuscript considers elliptic Dedekind sums introduced by Sczech as generalizations of the classical Dedekind sums to complex lattices. It proves that these sums, when suitably normalized, have a Gaussian limiting distribution. As an application, the paper proves a conjecture due to Ito.

Significance. If the central result holds, it would extend the study of limiting distributions for arithmetic sums attached to lattices, furnishing a central limit theorem in this elliptic setting and resolving an open conjecture. The work builds on Sczech's construction and applies probabilistic or analytic tools to obtain the Gaussian limit.

major comments (2)
  1. [Main theorem statement and preceding definitions] The normalization factor for the elliptic Dedekind sums is described as 'suitable' in the statement of the main result, but no explicit derivation from a second-moment calculation is supplied to confirm that the scaling yields finite, nonzero variance independently of the choice. Without this, it is unclear whether the Gaussian limit is a genuine theorem about the intrinsic distribution or follows tautologically from forcing the variance to 1. This is load-bearing for the central claim.
  2. [Application section] The application proving Ito's conjecture relies on the limiting distribution result. If the normalization step requires additional justification, the proof of the conjecture inherits the same gap and should be checked for independence from the scaling choice.
minor comments (2)
  1. [Abstract] The abstract supplies no proof sketch, error bounds, or outline of the analytic machinery (e.g., characteristic functions or moment-generating functions) used to establish the Gaussian limit; adding a brief indication in the introduction would improve readability.
  2. [Preliminaries] Notation for the lattice parameters and the precise form of the normalization constant should be introduced with explicit cross-references to the relevant equations in Sczech's construction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the constructive comments on the normalization of the elliptic Dedekind sums. We address the major comments point by point below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

  1. Referee: [Main theorem statement and preceding definitions] The normalization factor for the elliptic Dedekind sums is described as 'suitable' in the statement of the main result, but no explicit derivation from a second-moment calculation is supplied to confirm that the scaling yields finite, nonzero variance independently of the choice. Without this, it is unclear whether the Gaussian limit is a genuine theorem about the intrinsic distribution or follows tautologically from forcing the variance to 1. This is load-bearing for the central claim.

    Authors: We agree that an explicit second-moment calculation is needed to derive the normalization factor rigorously and to confirm that the resulting variance is finite, positive, and independent of the lattice in the appropriate asymptotic regime. In the revised manuscript we will insert a dedicated subsection (immediately preceding the statement of the main theorem) that computes the asymptotic variance of the un-normalized elliptic Dedekind sums via the explicit formulas of Sczech and standard estimates on the associated Eisenstein series. This calculation will show that the variance grows linearly with the summation parameter, yielding a positive constant that depends only on the lattice class; the normalization is then taken to be the square root of this variance. The subsequent proof of convergence in distribution proceeds by the method of characteristic functions and is independent of the particular scaling once the variance is fixed to 1. We do not regard the present argument as tautological, but we acknowledge that the explicit variance derivation was omitted and will be supplied in the revision. revision: yes

  2. Referee: [Application section] The application proving Ito's conjecture relies on the limiting distribution result. If the normalization step requires additional justification, the proof of the conjecture inherits the same gap and should be checked for independence from the scaling choice.

    Authors: The proof of Ito's conjecture is obtained by specializing the main limiting-distribution theorem to a particular sequence of lattices and summation parameters. Consequently, once the normalization is justified by the second-moment calculation in the main theorem, the application inherits the same justification and is independent of arbitrary rescaling. In the revised version we will add a short paragraph in the application section that explicitly references the variance computation and notes that the conjecture is recovered precisely when the sums are normalized to have asymptotic variance 1. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in the derivation chain.

full rationale

The paper defines elliptic Dedekind sums via Sczech's construction on complex lattices, states a normalization, and claims to prove a Gaussian limiting distribution (plus an application to Ito's conjecture). No quoted equations or sections exhibit a self-definitional reduction, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain that collapses the central claim to its inputs by construction. The normalization is described as 'suitable' for the limit to exist, which is standard for CLT-type statements once variance is computed independently; the proof itself is presented as relying on external analytic or probabilistic tools rather than tautological re-labeling. This is the expected non-finding for a self-contained analytic number-theory result.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be identified from the given text.

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