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REAL ZEROS OF L-FUNCTION DERIVATIVES: DENSITY-ONE ASYMPTOTICS IN ARITHMETIC FAMILIES

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theorymath.FA — Functional Analysis

Contributed by Yuhyun Han ↗

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

We study real zeros of the derivative of an uncompleted L-function on [1/2, 1]. The limiting stationary process has covariance sech2((t− s)/2) after the change of variables σ = 1/2+e^−t. Its nonzero-level Rice intensity gives the constant e^−1/2/(π√2) for quadratic Dirichlet characters, a specified family of quadratic Hecke characters over Q(i), and the unweighted level-one holomorphic family; the resulting density-one asymptotics are unconditional. For a fixed holomorphic modular form and its quadratic twists, and for non-Galois cubic Artin L- functions, we obtain the same constant under the respective family GRH. The real cyclic cubic augmentation instead has constant 1/(π√2). In these three GRH families a uniform quantile estimate controls all three counts: distinct real zeros, zeros with multiplicity, and strict sign changes. For an individual complex cubic character, the derivative has o(log log q) real zeros in density under cubic-character GRH. We also prove a uniform mean asymptotic and exponential concentration for the random Euler model.

Provenance statement

This project began in the first week of September 2026. Its starting points were the work of Klurman–Lamzouri–Munsch (KLM) and Lamzouri–Nath (LN), together with a lecture by Youness Lamzouri that I attended in person in Incheon in late August 2026. Building on and synthesizing ideas and results from KLM and LN, the initial phase developed an asymptotic formula for the number of real zeros of L′(s, χ_d) in the quadratic Dirichlet setting, refining the order-of-magnitude prediction of the Baker–Montgomery conjecture. This initial phase involved approximately 100 hours of work with ChatGPT, primarily using Astra in Ultra mode across three ChatGPT accounts, supplemented by Claude Fable 5.1 in Max mode. I directed an iterative process of mathematical development and critical review, repeatedly transferring arguments and drafts between separate sessions and accounts. The systems were asked to challenge previous outputs, identify gaps and unjustified assumptions, rederive mathematical steps, and assess proposed corrections. These early interactions took place outside the account used for the subsequent work. Once the initial asymptotic formula had been obtained, I consolidated the project in a single Codex working environment and used it to develop extensions of that result. My supervisor’s suggestion to investigate broader classes of L-functions, particularly Artin L-functions, guided this stage. The subsequent workflow included further proof development, comparison with original research papers, separate AI review sessions, numerical diagnostics, and manuscript preparation. Python was used for numerical diagnostics and consistency checks, and LaTeX for preparing the manuscript. AI systems contributed substantially to the mathematical exploration, generation and revision of arguments, and writing of the manuscript. My role was to select and refine the questions and target families, set research priorities, organize and direct repeated critical reviews, and decide how to revise and extend the work. Verification was extensively AI-assisted; these internal reviews do not constitute external peer review or formal proof verification. I take responsibility for the submitted manuscript and for accurately describing how it was produced.

Tools used

OpenAI
CodexVersion 6
Anthropic
Claude FableVersion 5.1

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Version history

  1. v1Submitted by Yuhyun HanInitial depositCurrentNov 04, 2026