Size of \(L(1,\chi)\)

Collecting references: [Louboutin, 1993],

1. Upper bounds for \(|L(1,\chi)|\)

[Louboutin, 1996], [Louboutin, 1998], [Granville and Soundararajan, 2003], [Granville and Soundararajan, 2004]. [Ramaré, 2001],

Theorem (2001)

For any primitive Dirichlet character \(\chi\) of conductor \(q\), we have \(\displaystyle |L(1,\chi)|\le\tfrac12\log q +\begin{cases}0&\text{when }\chi\text{ is even (i.e. }\chi(-1)=1\text{)},\\\tfrac52-\log 6&\text{when }\chi\text{ is odd (i.e. }\chi(-1)=-1\text{)}.\end{cases}\)

When the conductor \(q\) is even, this may be improved.

Theorem (2001)

For any primitive Dirichlet character \(\chi\) of even conductor \(q\), we have \(\displaystyle |L(1,\chi)|\le\tfrac14\log q+\begin{cases}\tfrac12\log 2&\text{when }\chi\text{ is even (i.e. }\chi(-1)=1\text{)},\\\tfrac54-\tfrac12\log 3&\text{when }\chi\text{ is odd (i.e. }\chi(-1)=-1\text{)}.\end{cases}\)

Similar bounds or more precise bounds may be found in [Louboutin, 2002], in [Ramaré, 2004] and in [Louboutin, 2018],

In [Platt and Eddin, 2013], we find the next result.

Theorem (2013)

For any primitive Dirichlet character \(\chi\) of conductor \(q\) divisible by 3, we have \(\displaystyle |L(1,\chi)|\le\tfrac13\log q+\begin{cases}0.368296&\text{when }\chi\text{ is even (i.e. }\chi(-1)=1\text{)},\\0.838374&\text{when }\chi\text{ is odd (i.e. }\chi(-1)=-1\text{)}.\end{cases}\)

In [Saad Eddin, 2016], we find improvement on the \((1/2)\log q\) bound in a very special (and difficult) case.

Theorem (2016)

For any primitive Dirichlet even character \(\chi\) of conductor \(q\) and with \(\chi(2)=1\), we have \(|L(1,\chi)|\le\tfrac12\log q - 0.02012\).

The general upper bounds are improved in [Johnston et al., 2023] as follows.

Theorem (2023)

For any quadratic primitive Dirichlet character \(\chi\) of conductor \(f\ge 2\cdot 10^{23}\), we have \(|L(1,\chi)|\le (1/ 2) \log f\).

Theorem (2023)

For any quadratic primitive Dirichlet character \(\chi\) of conductor \(f\ge 5\cdot 10^{50}\), we have \(|L(1,\chi)|\le (9/ 20) \log f\). When \(f\) is even, the lower bound on \(f\) may be improved to \(f\ge 2\cdot 10^{49}\).

2. Lower bounds for \(|L(1,\chi)|\)

[Louboutin, 2013] announces the following lower bound proved in [Louboutin, 2015].

Theorem (2013)

For any non-quadratic primitive Dirichlet character \(\chi\) of conductor \(f\), we have \(|L(1,\chi)|\ge 1/ ( 10\log(f/\pi))\).

This is improved in [Mossinghoff et al., 2022] where we find the next bound.

Theorem (2022)

For any non-quadratic primitive Dirichlet character \(\chi\) of conductor \(f\), we have \(|L(1,\chi)|\ge 1/ ( 9.69030\log(f/\pi))\).