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arXiv · 2511.08783

Towards Keating-Snaith's conjecture for cubic Hecke $L$-functions over the Eisenstein field

Hua Lin·Peng-Jie Wong
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Abstract

A famous conjecture of Keating and Snaith asserts that central values of $L$-functions in a given family admit a log-normal distribution with a prescribed mean and variance depending on the symmetry type of the family. Based on a recent work of Radziwill and Soundararajan, we obtain a conditional lower bound towards Keating-Snaith's conjecture for a "thin" family of cubic Hecke $L$-functions over the Eisenstein field. A key new input is certain twisted estimates of the 1-level density of zeros of cubic Hecke $L$-functions, extending the previous work of David and G\"{u}lo\u{g}lu, under the Generalised Riemann Hypothesis.

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Hua Lin, Peng-Jie Wong. 2025-11-11. Towards Keating-Snaith's conjecture for cubic Hecke $L$-functions over the Eisenstein field. https://arxiv.org/abs/2511.08783

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