Uniform effective estimates for $\vert L(1,χ)\vert$

TL;DR

The paper addresses obtaining uniform, explicit GRH-conditional bounds for |L(1, χ)| across all moduli q, including composite ones. It adapts and extends the Lamzouri–Li–Soundararajan inequalities by refining the choice of truncation parameter and error analysis, proving upper and lower bounds for q ≥ 404 and then validating them computationally for 3 ≤ q ≤ 1000. A key consequence is new bounds for the class numbers h(-q) of imaginary quadratic fields via Dirichlet’s class number formula, extending previous ranges to q ≥ 5. Overall, the work provides uniform, effective estimates for |L(1, χ)| and translates them into arithmetic consequences for imaginary quadratic fields, supported by substantial computational verification.

Abstract

Let $L(s,χ)$ be the Dirichlet $L$-function associated to a non-principal primitive Dirichlet character $χ$ defined modulo $q$, where $q\ge 3$. We prove, under the assumption of the Generalised Riemann Hypothesis, the validity of estimates given by Lamzouri, Li, and Soundararajan on $\vert L(1,χ) \vert$. As a corollary, we have that similar estimates hold for the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-q})$, $q\ge 5$.

Figures (6)

• Figure 1: The values of $M_q$, $3\le q\le 1000$. The minimal value for $M_q$ is $0.604599\dotsc$ attained at $q=3$ and the maximal one is $3.652103\dotsc$ attained at $q= 959$. The blue line represents $0.47\cdot L_2f(q)$, $L_2= 2 e^\gamma$, where $f(q)$ is defined in \ref{['fg-def']}. The green line represents $0.71\cdot L_2\log \log q$.
• Figure 2: The values of $M_q^\prime:=M_q/f(q)$, $3\le q\le 1000$, where $f(q)$ is defined in \ref{['fg-def']}. The minimal value for $M_q^\prime$ is $0.057396\dotsc$ attained at $q=3$ and the maximal one is $1.648945\dotsc$ attained at $q= 479$. The blue line represents $0.47\cdot L_2$, $L_2= 2 e^\gamma$.
• Figure 3: The values of $M_q^{\prime\prime}:=M_q/\log \log q$, $3\le q\le 1000$. The minimal value for $M_q^{\prime\prime}$ is $0.799902\dotsc$ attained at $q=144$ and the maximal one is $6.428641 \dotsc$ attained at $q= 3$. The green line represents $0.71\cdot L_2$, where $L_2= 2 e^\gamma$.
• Figure 4: The values of $m_q$, $3\le q\le 1000$. The minimal value for $m_q$ is $0.246068\dotsc$ attained at $q=163$ and the maximal one is $0.785398\dotsc$ attained at $q=4$. The red line represents $3.25 L_1 / g(q)$, where $g(q)$ is defined in \ref{['fg-def']} and $L_1= \frac{\pi^2}{12 e^\gamma}$. The orange line represents $0.425 L_1/\log\log q$.
• Figure 5: The values of $m_q^\prime:=m_q g(q)$, $3\le q\le 1000$, where $g(q)$ is defined in \ref{['fg-def']}. The minimal value for $m_q^\prime$ is $1.520104\dotsc$ attained at $q=978$ and the maximal one is $7.093329\dotsc$ attained at $q=3$. The red line represents $3.25 L_1$, where $L_1= \frac{\pi^2}{12 e^\gamma}$.

Theorems & Definitions (7)

• Theorem 1
• Corollary 1
• Lemma 1: Lemma 2.3 of LamzouriLS2015
• Lemma 2: Lemma 2.4 of LamzouriLS2015
• Lemma 3: Lemma 2.5 of LamzouriLS2015
• Lemma 4: Lemma 2.6 of LamzouriLS2015
• Lemma 5: Lemma 5.1 of LamzouriLS2015