Record W4411567554 · doi:10.5802/jtnb.1318

Large Sums of Fourier Coefficients of Cusp Forms

2025· article· lv· W4411567554 on OpenAlexfundno aff

Claire Frechette, Mathilde Gerbelli-Gauthier, Alia Hamieh, Naomi Tanabe

Bibliographic record

VenueJournal de Théorie des Nombres de Bordeaux · 2025

Typearticle

Languagelv

FieldMathematics

TopicAnalytic Number Theory Research

Canadian institutionsnot available

FundersBanff International Research Station for Mathematical Innovation and Discovery

KeywordsComputer science

DOI10.5802/jtnb.1318

Abstract

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Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> be a fixed positive integer, and let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> be a primitive cusp form given by the Fourier expansion <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>∞</mml:mi> </mml:msubsup> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:msup> <mml:mi>n</mml:mi> <mml:mfrac> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mn>2</mml:mn> </mml:mfrac> </mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . We consider the partial sum <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>f</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . It is conjectured that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>f</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo form="prefix">log</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> in the range <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>≥</mml:mo> <mml:msup> <mml:mi>k</mml:mi> <mml:mi>ϵ</mml:mi> </mml:msup> </mml:mrow> </mml:math> . Lamzouri proved in [8] that this is true under the assumption of the Generalized Riemann Hypothesis (GRH) for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>f</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . In this paper, we prove that this conjecture holds under a weaker assumption than GRH. In particular, we prove that given <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϵ</mml:mi> <mml:mo>></mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo form="prefix">log</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>8</mml:mn> </mml:mfrac> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>T</mml:mi> <mml:mo>≤</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo form="prefix">log</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>200</mml:mn> </mml:mfrac> </mml:msup> </mml:mrow> </mml:math> , we have <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>f</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>≪</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo form="prefix">log</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> <mml:mi>T</mml:mi> </mml:mfrac> </mml:mrow> </mml:math> in the range <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>≥</mml:mo> <mml:msup> <mml:mi>k</mml:mi> <mml:mi>ϵ</mml:mi> </mml:msup> </mml:mrow> </mml:math> provided that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>f</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> has no more than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">

Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.

How this classification was reachedexpand

Full frame distilled prediction

Teacher imitation

Not calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.

metaresearch head score (Codex)0.006

metaresearch head score (Gemma)0.003

Version: codex-gemma-dda1882f352aValidation status: machine_predicted_unvalidated

Candidate categoriesMeta-epidemiology (narrow), Insufficient payload (model declined to judge)

Consensus categoriesnone

DomainCandidate signal: none · Consensus signal: none

Study designCandidate signal: Theoretical or conceptual · Consensus signal: Theoretical or conceptual

GenreCandidate signal: Empirical · Consensus signal: Empirical

Teacher disagreement score0.205

Threshold uncertainty score1.000

Codex and Gemma teacher scores by category

Category	Codex	Gemma
Metaresearch	0.006	0.003
Meta-epidemiology (narrow)	0.000	0.000
Meta-epidemiology (broad)	0.001	0.001
Bibliometrics	0.001	0.001
Science and technology studies	0.000	0.001
Scholarly communication	0.000	0.000
Open science	0.001	0.001
Research integrity	0.000	0.001
Insufficient payload (model declined to judge)	0.001	0.000

Machine scores (provisional)

The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.

Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.

Opus teacher head0.023

GPT teacher head0.346

Teacher spread0.324 · how far apart the two teachers sit on this one work

Validation statusscore_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it

Classification

machine, unvalidated

Machine predicted; a candidate call from one teacher head, not a consensus.

Meta-epidemiology (narrow)Insufficient payload (model declined to judge)

Study designTheoretical or conceptual

Domainnot available

GenreEmpirical

How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".

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Citations0

Published2025

Admission routes1

Has abstractyes

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