Pieter Moree: Euler-Kronecker constants: from Ramanujan to Ihara

 Given an $L$-series that has a series expansion around $s=1$ starting as $c_1(s-1)^{\alpha}+c_2+\ldots$, one can define its Euler-Kronecker constant as $c_2/c_1$. Ramanujan in his `unpublished' manuscript on the Ramanujan tau-function made various conjectures on Euler-Kronecker constants. In case the $L$-series is the Dedekind zeta function, $\zeta_K(s)$, of a number field $K$ (in which case $\alpha=-1$), this constant has been intensively studied by Ihara (of the Ihara zeta function) and his collaborators. We show amongst others that, assuming some widely believed conjectures, a conjecture made by Ihara in case $K={\Bbb Q}(\zeta_q)$ is a cyclotomic field, $q$ a prime, is false. We also deal with the analogue of Ramanujan's conjectures for these fields. (Joint work with Kevin Ford and Florian Luca, to appear in Mathematics of Computation)