Extremal values of Dirichlet \(L\)-functions in the half-plane of absolute convergence
Let \(q\) be a positive integer and let \(\chi\) be a Dirichlet character mod \(q\). As usual, denote by \[ L(s, \chi) ~=~ \sum_{n=1}^{\infty} \frac{\chi (n)}{n^s} ~=~ \prod_{p} \left( 1 - \frac{\chi (p)}{p^s} \right)^{-1} \] the Dirichlet L-function associated to \(\chi\). The author proves that for any real \(\theta\) there are infinitely many values of \(s= \sigma + it\) with \(\sigma \to 1+\) and \(t \to \infty\) such that \[ Re \left \{ \exp(i \theta) \log L(s, \chi) \right \} ~\geq~ \log \frac{\log \log \log t}{\log \log \log \log t} ~+~ O(1) ~~. \] The proof is based on an effective version of Kronecker's approximation theorem.
scientific article; zbMATH DE number 2184642