Refined Upper Bounds for $L(1,\chi )$

Let $\chi$ be a non-principal Dirichlet character of modulus $q$ with associated $\mathit{\text{L}}$-function $L(s,\chi )$. We prove that $$\mathrm{\mid }L(1,\chi )\mathrm{\mid }\le (\frac{1}{2}+O(\frac{\log \log q}{\log q}\left)\right)\frac{\phi (q)}{q}\log q,$$ where $\phi (\cdot )$ is Euler’s phi function. This refines known bounds of the form $(c+o(1))\log q$ or $(c+O(\frac{1}{\log q}))\log q$ and is relevant for prime-rich moduli. It follows from Mertens’ third theorem and the prime number theorem that ${\inf }_{q>2}{\max }_{\chi \mathrm{\ne }{\chi }_0(\mathrm{m}\mathrm{o}\mathrm{d}q)}\frac{\mathrm{\mid }L(1,\chi )\mathrm{\mid }}{\log q\mathrm{/}\log \log q}\le \frac{1}{2}{e}^{-\gamma }$.