Euler-Kronecker constants for cyclotomic fields

TL;DR

This paper investigates the size and distribution of Euler-Kronecker constants for cyclotomic fields, showing under the Elliott-Halberstam Conjecture that these constants normalized by log q converge to 1 in average.

Contribution

It extends previous results by proving uniform convergence of the normalized Euler-Kronecker constants for cyclotomic fields under EH, generalizing from prime to composite q.

Findings

Under EH, the average difference between γ_q and log q tends to zero.

The normalized γ_q / log q converges to 1 in distribution.

The result generalizes prior work from prime to all q in the specified range.

Abstract

The Euler-Mascheroni constant $\gamma=0.5772\dots\!$ is the $K=\mathbb{Q}$ example of an Euler-Kronecker constant $\gamma_K$ of a number field $K.$ In this note we consider the size of the $\gamma_q=\gamma_{K_q}$ for cyclotomic fields $K_q:=\mathbb{Q}(\zeta_q).$ Assuming the Elliott-Halberstam Conjecture (EH), we prove uniformly in $Q$ that $$\frac{1}{Q}\sum_{Q<q\le 2Q} \left |\gamma_q - \log q\right |= o(\log Q).$$ In other words, under EH the $\gamma_q / \log q$ in these ranges converge to the one point distribution at $1$. This theorem refines and extends a previous result of Ford, Luca, and Moree for prime $q.$