Numerical estimates on the Landau-Siegel zero and other related quantities

TL;DR

The paper conducts a comprehensive numerical investigation of Landau–Siegel zeros for the family of quadratic Dirichlet L-functions $L(s,\chi_{\square})$ with $\chi_{\square}$ mod prime $q \le 10^7$. It develops and implements a fast FFT-based approach to compute $|L(1,\chi_{\square})|$ with high precision, enabling explicit bounds that relate potential Landau–Siegel zeros to $L(1,\chi_{\square})$ via $L(1,\chi_{\square}) > c_1 \log q$ and $\beta < 1-\frac{c_2}{\log q}$. The work provides computable constants $c_1,c_2>0$ (with $c_1=0.0124862668\dots$, $c_2=0.0091904477\dots$) and reports extremal values across primes, situating the results within Littlewood/Joshi bounds and class-number theory; extensive data and methodological details, including FFT accuracy analyses, are made publicly available for reproducibility. The combination of explicit inequalities, high-precision L-values, and large-scale computation advances understanding of Siegel zeros in this setting and offers practical numerical bounds for related arithmetic quantities such as class numbers $h(-q)$. The approach showcases a scalable framework for numerically probing bias and zeros in families of Dirichlet L-functions.

Abstract

Let $q$ be a prime, $χ$ be a non-principal Dirichlet character $\bmod\ q$ and $L(s,χ)$ be the associated Dirichlet $L$-function. For every odd prime $q\le 10^7$, we show that $L(1,χ_\square) > c_{1} \log q$ and $β< 1- \frac{c_{2}}{\log q}$, where $c_1=0.0124862668\dotsc$, $c_2=0.0091904477\dotsc$, $χ_{\square}$ is the quadratic Dirichlet character $\bmod\ q$ and $β\in (0,1)$ is the Landau-Siegel zero, if it exists, of such a set of Dirichlet $L$-functions. As a by-product of the computations here performed, we also obtained some information about the Littlewood and Joshi bounds on $L(1,χ_\square)$ and on the class number of the imaginary quadratic field ${\mathbb Q}(\sqrt{-q})$.

Figures (10)

• Figure 1: The values of $c_1(q)$, $q$ prime, $3\le q\le 10^7$, in Theorem \ref{['Thm-beta']}. $\min c_{1}(q) = 0.0124862668\dotsc$ attained at $q=7105733$; $\max c_{1}(q) = 0.6267599041\dotsc$ attained at $q=23$. The black straight line corresponds to $0.627$; the red one to $0.012$. The black line corresponds to the first Joshi bound, see \ref{['Joshi-first']}, the yellow one corresponds to the second Joshi bound, see \ref{['Joshi-second']}. The red dashed line corresponds to the mean value.
• Figure 2: The values of $0.0124<c_1(q)\le 0.022$, $q$ prime, $3\le q\le 10^7$, in Theorem \ref{['Thm-beta']}. $\min c_{1}(q) = 0.0124862668\dotsc$ attained at $q=7105733$. The red straight lines correspond to $0.0124$ and $0.022$. The yellow one corresponds to the second Joshi bound, see \ref{['Joshi-second']}; the black one corresponds to the second Littlewood bound, see \ref{['Littlewood-bounds']}.
• Figure 3: The values of $c_2(q)$, $q$ prime, $3\le q\le 10^7$, in Theorem \ref{['Thm-beta']}. $\min c_{2}(q) = 0.0091904477\dotsc$ attained at $q=7105733$; $\max c_{2}(q) = 0.4206022969\dotsc$ attained at $q=311$. The black line corresponds to $0.42$; the red one to $0.009$. The red dashed line corresponds to the mean value.
• Figure 4: The values of $0.00915<c_2(q) \le 0.012$, $q$ prime, $3\le q\le 10^7$, in Theorem \ref{['Thm-beta']}. $\min c_{2}(q) = 0.0091904477\dotsc$ attained at $q=7105733$. The red lines correspond to $0.00915$ and $0.012$.
• Figure 5: Histogram about the values of $c_1(q)$, $q$ prime, $3\le q\le 10^7$, in Theorem \ref{['Thm-beta']}. Intervals length $:= I = 0.0067570100\dotsc$; number of primes $3\le q\le 10^7 := {\mathcal{P}} = 664578$; mass $:= \mathcal{M} = I \cdot {\mathcal{P}}$; mean $:= \mu = 0.1096877373\dotsc$; standard deviation $:= \sigma =0.0788767546\dotsc$ The red dashed line corresponds to the mean value.

Theorems & Definitions (4)

• Theorem 1
• Lemma 1
• Lemma 2
• Lemma 3