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 $q.$

Accepted version of the paper. To appear in the Bulletin of the Australian Mathematical Society