The TME-EMT project
Size of $L(1,\chi)$
Collecting references:
,
1. Upper bounds for $|L(1,\chi)|$
,
,
,
.
,
Theorem (2001)
For any primitive Dirichlet character $\chi$ of conductor $q$,
we have
$$
|L(1,\chi)|\le\tfrac12\log q
+
\begin{cases}
0&\text{when $\chi$ is even (i.e. $\chi(-1)=1$),}\\
\tfrac52-\log 6&\text{when $\chi$ is odd (i.e. $\chi(-1)=-1$).}
\end{cases}
$$
When the conductor $q$ is even, this may be improved.
Theorem (2001)
For any primitive Dirichlet character $\chi$ of even conductor $q$,
we have
$$
|L(1,\chi)|\le\tfrac14\log q
+
\begin{cases}
\tfrac12\log 2&\text{when $\chi$ is even (i.e. $\chi(-1)=1$),}\\
\tfrac54-\tfrac12\log 3&\text{when $\chi$ is odd (i.e. $\chi(-1)=-1$).}
\end{cases}
$$
Similar bounds or more precise bounds may be found in
,
in
and in
,
In
,
we find the next result.
Theorem (2013)
For any primitive Dirichlet character $\chi$ of conductor $q$
divisible by 3,
we have
$$
|L(1,\chi)|\le\tfrac13\log q
+
\begin{cases}
0.368296&\text{when $\chi$ is even (i.e. $\chi(-1)=1$),}\\
0.838374&\text{when $\chi$ is odd (i.e. $\chi(-1)=-1$).}
\end{cases}
$$
In
,
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 $q$,
we have
$ |L(1,\chi)|\le\tfrac12\log q - 0.02012 $.
The general upper bounds are improved in
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 $|L(1,\chi)|$
announces the following lower bound proved in
.
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
where we find the next bound.
Theorem (2022)
For any non-quadratic primitive Dirichlet character $\chi$ of conductor $f$,
we have $|L(1,\chi)|\ge 1/ ( 9.69030\log(f/\pi))$.
Last updated on February 7th, 2024, by Olivier Ramaré