Asymmetric Distribution of Extreme Values of Cubic $L$-functions at $s=1$

TL;DR

This paper studies the distribution of cubic Dirichlet L-functions at s=1, revealing an asymmetry in the likelihood of small versus large values, and models this distribution using random Euler products.

Contribution

It extends the understanding of L-function value distributions to cubic characters, highlighting asymmetries not present in quadratic cases, and employs probabilistic models for analysis.

Findings

Small values of |L(1,χ)| are less probable than large values.

The distribution exhibits asymmetry between lower and upper bounds.

Provides a probabilistic description of the distribution of L-values.

Abstract

We investigate the distribution of values of cubic Dirichlet $L$-functions at $s=1$. Following ideas of Granville and Soundararajan for quadratic $L$-functions, we model the distribution of $L(1,\chi)$ by the distribution of random Euler products $L(1,\mathbb{X})$ for certain family of random variables $\mathbb{X}(p)$ attached to each prime. We obtain a description of the proportion of $|L(1,\chi)|$ that are larger or that are smaller than a given bound, and yield more light into the Littlewood bounds. Unlike the quadratic case, there is an asymmetry between lower and upper bounds for the cubic case, and small values are less probable than large values.