Global Fields

July 2 - 6, 2007, Moscow, Russia

Laboratoire J.-V. Poncelet

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Andrey Badzjan

Independent University of Moscow, Russia

On the Euler-Kronecker constant

Let K be a global field. Let $\zeta_K(s)$ be the Dedikind zeta function of K, with the Laurent expansion at $s=1, \zeta_K(s)=c_{-1}(s-1)^{-1}+c_0+c_1(s-1)+...$. The real number $\gamma_K = c_0/c_{-1}$ is called the Euler-Kronecker constant of the field $K$. In the article of Yasutaka Ihara "On the Euler-Kronecker constants of global fields and primes with small norms" the lower bound $\gamma_K\ge-\log\sqrt{|d|}$ is obtained, where $d$ is the discriminant of the field $K$. In my talk I will prove the lower bound $\gamma_K\ge -\left(1-\frac{\sqrt{5}}{5}\right) \log\sqrt{|d|}$. Also, assuming the generalized Riemann hypothesis the bound $\gamma_K\ge -0.459\log\sqrt{|d|}$ will be proven.


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