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

Let $\chi$ be a non-principal Dirichlet character of modulus $q$ with associated \textit{L}-function $L(s,\chi)$. We prove that $$|L(1,\chi)|\le\left(\frac{1}{2}+O\Big(\frac{\log\log q}{\log q}\Big)\right)\frac{\varphi(q)}{q}\log q\,,$$ where $\varphi(\cdot)$ 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 $\inf_{q>2}\max_{\chi\ne\chi_0\,(\mod q)}\frac{|L(1,\chi)|}{\log q/\log\log q}\le\frac{1}{2}e^{-\gamma}$.