Size of :math:`L(1,\chi)`
=========================

.. if-builder:: html

   .. toctree::
      :maxdepth: 2




Collecting references:
:cite:`Louboutin93`,


1. Upper bounds for :math:`|L(1,\chi)|`
---------------------------------------


:cite:`Louboutin96`,
:cite:`Louboutin98`,
:cite:`Granville-Soundararajan02`,
:cite:`Granville-Soundararajan04`.
:cite:`Ramare01`,



.. admonition:: Theorem (2001)
   :class: thm-tme-emt

   For any primitive Dirichlet character :math:`\chi` of conductor :math:`q`,
   we have
   :math:`\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 :math:`q` is even, this may be improved.

.. admonition:: Theorem (2001)
   :class: thm-tme-emt
 
   For any primitive Dirichlet character :math:`\chi` of even conductor :math:`q`,
   we have
   :math:`\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
:cite:`Louboutin02-a`,
in
:cite:`Ramare02-??`
and in
:cite:`Louboutin18`,

 
In
:cite:`Platt-Eddin13`,
we find the next result.

.. admonition:: Theorem (2013)
   :class: thm-tme-emt

   For any primitive Dirichlet character :math:`\chi` of conductor :math:`q`
   divisible by 3,
   we have
   :math:`\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
:cite:`SaadEddin16`,
we find improvement on the :math:`(1/2)\log q` bound in a very special (and
difficult) case.

.. admonition:: Theorem (2016)
   :class: thm-tme-emt

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



The general upper bounds are improved in
:cite:`Johnston-Ramare-Trudgian23`
as follows.


.. admonition:: Theorem (2023)
   :class: thm-tme-emt

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



.. admonition:: Theorem (2023)
   :class: thm-tme-emt

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




2. Lower bounds for :math:`|L(1,\chi)|`
---------------------------------------

:cite:`Louboutin13`
announces the following lower bound proved in
:cite:`Louboutin15`.


.. admonition:: Theorem (2013)
   :class: thm-tme-emt

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


This is improved in
:cite:`Mossinghoff-Starichkova-Trudgian22`
where we find the next bound.

.. admonition:: Theorem (2022)
   :class: thm-tme-emt

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