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The Euler-Kronecker constants of number fields

Banff International Research Station for Mathematical Innovation and Discovery

For a number field $K$, Ihara has introduced an invariant $\gamma_K$, called the Euler-Kronecker constant, which is closely related to the values of the logarithmic derivative of $L$-functions at $1$. More precisely, $$\gamma_K=\lim_{s\rightarrow 1^+} \left( \frac{\zeta_K^\prime(s)}{\zeta_K(s)}+ \frac{1}{s-1} \right),$$ where $\zeta_K(s)$ is the Dedekind zeta function of $K$. In past few years, the size and the sign of these constants have extensively been investigated for certain families of number fields. In this talk we outline Ihara's approach in a systematic study of these constants and as a sample result we describe our joint work with Alia Hamieh (UNBC) on the existence of a distribution function for the Euler-Kronecker constants of certain cubic extensions of $\mathbb{Q}(\sqrt{-3})$. Also as another example we describe the role these constants played in our recent joint work with Forrest Francis (UNSW Canberra) regarding the inequalities involving Euler's function that are equivalent to GRH.

Mathematics; Number theory; Combinatorics; Arithmetic number theory

video/mp4

Author affiliation: University of Lethbridge

2019-11-09

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/72235

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/