Euler-Kronecker constants for cyclotomic fields

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 $q.$