On Ihara’s conjectures for Euler–Kronecker constants
Volume 210 / 2023
Acta Arithmetica 210 (2023), 95-123
MSC: Primary 11R42; Secondary 11M20, 11R44, 11N13, 11M36.
DOI: 10.4064/aa220729-19-3
Published online: 4 May 2023
Abstract
As a natural generalization of the Euler–Mascheroni constant $\gamma $, Y. Ihara introduced the Euler–Kronecker constant $\gamma _K$ attached to any number field $K$. He obtained bounds on $\gamma _K$ conditional upon the generalized Riemann hypothesis. In this paper, we establish unconditional bounds on $\gamma _K$ in terms of the Siegel zero of $\zeta _K(s)$. We also produce an alternative proof of Ihara’s theorem without invoking the explicit formula. Furthermore, using known upper bounds on $\gamma _{\mathbb Q(\zeta _q)}$, we obtain a bound on the error term in the prime number theorem, averaging over certain arithmetic progressions.
