Refined Upper Bounds for $L(1,χ)$
Jeffery Ezearn
TL;DR
This paper derives a refined uniform upper bound for $L(1,\chi)$ for non-principal Dirichlet characters modulo $q$, namely $|L(1,\chi)|\le\left(\frac{1}{2}+O\left(\frac{\log\log q}{\log q}\right)\right)\frac{\varphi(q)}{q}\log q$, and shows $\inf_{q>2}\max_{\chi\ne\chi_0}\frac{|L(1,\chi)|}{\log q/\log\log q}\le\tfrac{1}{2}e^{-\gamma}$. The approach combines a simple lemma with classical estimates such as the Pólya–Vinogradov inequality, Mertens' theorems, and the prime number theorem, and also provides a digamma-function based argument yielding a related explicit bound. The results improve prior $(c+o(1))\log q$ bounds and extend to prime-rich moduli, highlighting the dependence on the prime factors of the modulus. Altogether, the work sharpens understanding of $L$-values at $s=1$ in terms of modulus arithmetic and primes.
Abstract
Let $χ$ be a non-principal Dirichlet character of modulus $q$ with associated \textit{L}-function $L(s,χ)$. We prove that $$|L(1,χ)|\le\left(\frac{1}{2}+O\Big(\frac{\log\log q}{\log q}\Big)\right)\frac{\varphi(q)}{q}\log q\,,$$ 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 $\inf_{q>2}\max_{χ\neχ_0\,(\mod q)}\frac{|L(1,χ)|}{\log q/\log\log q}\le\frac{1}{2}e^{-γ}$.