The Euler-Kronecker constant
Let \(K\) be an algebraic number field with Euler-Kronecker constant \(\gamma_K\) and discriminant \(d_K\). The author proves that \[ \gamma_K\geq -\left(1-\frac{\sqrt{5}}{5}\right)\log\sqrt{|d_K|}. \] If the generalized Riemann hypothesis is true, then \[ \gamma_K\geq -0.459 \log\sqrt{|d_K|} \] and \[ \gamma_{{\mathbb Q}(\mu_l)}=O(\log l\log\log l) \] for the cyclotomic fields \({\mathbb Q}(\mu_l)\). Suppose that \(X\) is a smooth projective manifold of dimension \(n\) over a finite field \({\mathbb F}_q\), and \(\zeta(s,X)\) is its zeta function with the Laurent expansion at the point \(s=n\): \[ \zeta(s,X)=c_{-1}(s-n)^{-1}+c_0+\dots. \] The quantity \( \gamma_X=\frac{c_0}{c_{-1}}\) is called the Euler-Kronecker constant of the manifold \(X\). Let \(b_i=b_i(X)\) be the Betti numbers of \(X\), \(i=0,1,\dots,2n\), and let \[ b(X):=\max_{i=0,1,\dots,2n}b_i. \] The author proves that \[ \gamma_X\geq \log q\left(\frac12-\sum_{i=0}^{n-1}\frac{b_{2i}}{q^{n-i}-1}- \sum_{i=1}^{n}\frac{b_{2i-1}}{q^{n+\frac12-i}+1}\right) \] and that \[ \gamma_X\leq \frac{\log q}{2}+\frac{q^n+1}{q^n-q^{n-\frac12}}\log q+2\log(b(X)+1). \]
scientific article; zbMATH DE number 5791017