Extreme values of Euler-Kronecker constants
A number field \(K\) of degree \(n\) and discriminant \(d_K\) is called an \(S_n\)-field if its Galois closure over \(\mathbb{Q}\) is an \(S_n\)-Galois extension. Let \(\zeta_K(s)\) be the Dedekind zeta function. Therefore, the Laurent series expansion of \(\frac{\zeta_K'(s)}{\zeta_K(s)}\) at \(s = 1\) is \[\frac{\zeta_K'(s)}{\zeta_K(s)}=\frac{1}{s-1}+\gamma_K+b_1(s-1)+\cdots,\] where \(\gamma_K\) is called the Euler-Kronecker constant of \(K\). In the paper under review, the author shows that assuming the Artin conjecture (i.e., \(\frac{\zeta_K'(s)}{\zeta(s)}\) is analytic) for \(S_n\)-fields with \(n\leq5\) and \(X/2\leq d_K\leq X\) except for a density zero set as \(X\to \infty\), the following lower and upper bound for \(\gamma_K\): \[ -(n-1) \log \log d_K+O(\log \log \log d_K)\leq \gamma_K\leq \log \log d_K+ O(\log \log \log d_K).\] It is also shown that that there exists an infinite family of \(S_n\)-fields with the above lower and upper bounds values.
logarithmic derivatives of \(L\)-functions
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