A purely analytical lower bound for \(L(1,\chi )\)
It is well-known that genus theory of quadratic forms and the Dirichlet class number formula gives the bound \(L(1,\chi)\geq \frac{2^{\omega(q)-1}\pi}{\sqrt{q}}\), where \(\omega(q)\) denotes the number of distinct prime factors of \(q\) and \(\chi\) is an odd primitive quadratic Dirichlet character of conductor \(q\). In the paper under review the author gives a simple and purely analytical proof of a lower bound of similar strength, namely he proves that \[ L(1,\chi)\geq \frac{2^{\omega(q)-1}\pi}{7.5\sqrt{q}}. \] Instead of the theory of quadratic forms and the Dirichlet class number formula, the author uses the functional equation of \(\Phi(s)=(2\pi/\sqrt{q})^{-s}\Gamma(s)\zeta(s)L(s,\chi)\) and some basic properties of the Dirichlet convolution \(\mu^2\star \chi\), where \(\mu\) denotes the Möbius function.
scientific article; zbMATH DE number 5629439
scientific article