Numerical verification of Littlewood's bounds for |L(1,χ)|

Abstract

Let $L(s,\chi )$ be the Dirichlet L-function associated to a non-principal primitive Dirichlet character χ defined $\mathrm{mod}q$, where q is an odd prime. In this paper we introduce a fast method to compute $|L(1,\chi )|$ using the values of Euler's Γ function. We also introduce an alternative way of computing $\log \mathrm{\Gamma }(x)$ and $\psi (x)={\mathrm{\Gamma }}^{\prime }/\mathrm{\Gamma }(x)$, $x\in (0,1)$. Using such algorithms we numerically verify the classical Littlewood bounds and the recent Lamzouri-Li-Soundararajan estimates on $|L(1,\chi )|$, where χ runs over the non-principal primitive Dirichlet characters $\mathrm{mod}q$, for every odd prime q up to 10⁷. The programs used and the results here described are collected at the following address http://www.math.unipd.it/~languasc/Littlewood_ineq.html.

Introduction

Let q be an odd prime, χ be a Dirichlet character $\mathrm{mod}q$ and $L(s,\chi )$ be the associated Dirichlet L-function. The goal of this paper is to introduce a fast algorithm to compute the values of $|L(1,\chi )|$ for every non-principal primitive Dirichlet character χ defined $\mathrm{mod}q$ and, using such a new method, to numerically study a generalisation of the classical bounds of Littlewood [12] for $|L(1,{\chi }_d)|$, where ${\chi }_d$ is a quadratic Dirichlet character. Assuming the Riemann Hypothesis for $L(s,{\chi }_d)$ holds, in 1928 Littlewood proved, for $d\ne m^2$, that

$${(\frac{12e^{\gamma }}{{\pi }^2}(1+o\left(1\right))\log \log d)}^{-1}<L(1,{\chi }_d)<2e^{\gamma }(1+o\left(1\right))\log \log d$$

as d tends to infinity, where γ is the Euler-Mascheroni constant. In 1973 Shanks [16] numerically studied the behaviour of the upper and lower Littlewood indices defined as

$$\text{ULI}(d,{\chi }_d):=\frac{L(1,{\chi }_d)}{2e^{\gamma }\log \log |d|}\text{and}\text{LLI}(d,{\chi }_d):=L(1,{\chi }_d)\frac{12e^{\gamma }}{{\pi }^2}\log \log |d|$$

for several small discriminants d. Such computations were extended by Williams-Broere [17] in 1976 and by Jacobson-Ramachandran-Williams [5] in 2006.

Recently Lamzouri-Li-Soundararajan [8, Theorem 1.5] proved an effective form of Littlewood's inequalities: assuming the Generalised Riemann Hypothesis holds, for every integer $q\ge {10}^{10}$ and for every non-principal primitive character $\chi \mathrm{mod}q$, they obtained that

$$|L(1,\chi )|\le 2e^{\gamma }(\log \log q-\log 2+\frac{1}{2}+\frac{1}{\log \log q})$$

and

$$\frac{1}{|L(1,\chi )|}\le \frac{12e^{\gamma }}{{\pi }^2}(\log \log q-\log 2+\frac{1}{2}+\frac{1}{\log \log q}+\frac{14\log \log q}{\log q}).$$

Using our method we will compute the values of $|L(1,\chi )|$ for every non-principal primitive Dirichlet character χ defined $\mathrm{mod}q$, for every odd prime q up to 10⁷. This largely extends previous results. Moreover, letting

$$M_q:=\underset{\chi \ne {\chi }_0}{\max }|L(1,\chi )|,m_q:=\underset{\chi \ne {\chi }_0}{\min }|L(1,\chi )|,$$

$$f(q):=\log \log q-\log 2+1/2+1/\log \log q,g(q):=f(q)+14(\log \log q)/\log q,$$

we obtain the following

Theorem 1

Let $3\le q\le {10}^7$, q be a prime number and $M_q$ be defined in (4). Then we have $0.604599\dots =M_3\le M_q\le M_{4305479}=6.399873\dots$. Moreover, we also have

$$0.325\cdot 2e^{\gamma }f(q)<M_q<0.62\cdot 2e^{\gamma }f(q),$$

where the lower bound holds just for $q\ge 79$, and

$$0.4<\underset{\chi \ne {\chi }_0}{\max }\mathit{\text{ULI}}(q,\chi )<0.66,$$

where the upper bound holds just for $q\ge 5$.

We also have an analogous result on $m_q$.

Theorem 2

Let $3\le q\le {10}^7$, q be a prime number and $m_q$ be defined in (4). Then we have $0.198814\dots =m_{991027}\le m_q\le m_{11}=0.618351\dots$. Moreover, we also have

$$\frac{{\pi }^2}{12e^{\gamma }}\frac{2.35}{g(q)}<m_q<\frac{{\pi }^2}{12e^{\gamma }}\frac{5}{g(q)},$$

where the upper bound holds just for $q\ge 953$, and

$$1.13<\underset{\chi \ne {\chi }_0}{\min }\mathit{\text{LLI}}(q,\chi )<2,$$

where the lower bound holds just for $q\ge 373$.

Theorem 1, Theorem 2 are in agreement with Littlewood's bounds in (1) and the Lamzouri-Li-Soundararajan estimates in (2)-(3).

The paper is organised as follows: in Section 2 we will see how to compute $|L(1,\chi )|$ using the values of Euler's Γ function and the Fast Fourier Transform algorithm; we will also describe the actual computation we performed and how Theorem 1, Theorem 2 are obtained. In Sections 3-4 we will see how to efficiently evaluate $\log \mathrm{\Gamma }(x)$, and $\psi (x)={\mathrm{\Gamma }}^{\prime }/\mathrm{\Gamma }(x)$, for $x\in (0,1)$ using precomputed values of the Riemann zeta-function at positive integers. We will also insert some tables and figures (the scatter plots were obtained using GNUPLOT, v.5.2.8).