Tools on Mellin transforms
==========================

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1. Explicit truncated Perron formula
------------------------------------



Here is Theorem 7.1 of
:cite:`Ramare07a`.


 
.. admonition:: Theorem (2007)
   :class: thm-tme-emt

   Let :math:`F(z)=\sum_{n}a_n/n^z` be a Dirichlet series that converges absolutely
   for :math:`\Re z>\kappa_a`, and let :math:`\kappa>0` be strictly larger than 
   :math:`\kappa_a`. For :math:`x\ge1` and :math:`T\ge1`, we have
   :math:`\displaystyle      \sum_{n\le x}a_n    =\frac1{2i\pi}\int_{\kappa-iT}^{\kappa+iT}F(z)\frac{x^zdz}z    +\mathcal{O}^*\left(      \int_{1/T}^{\infty}      \sum_{|\log(x/n)|\le u}\frac{|a_n|}{n^\kappa}      \frac{2x^\kappa du}{T u^2}    \right).` 


See
:cite:`Ramare14-6`
for different versions.




2. :math:`L{}^2`-means
-----------------------


We start with a majorant principle taken for instance from
:cite:`Montgomery94`,
chapter 7, Theorem 3.

 
.. admonition:: Theorem
   :class: thm-tme-emt

   Let :math:`\lambda_1,\cdots,\lambda_N` be :math:`N` real numbers, and suppose
   that :math:`|a_n|\le A_n` for all :math:`n`. Then
   :math:`\displaystyle    \int_{-T}^T\Bigl|\sum_{1\le n\le N}a_n e(\lambda_n t)\Bigr|^2dt  \le 3  \int_{-T}^T\Bigl|\sum_{1\le n\le N}A_n e(\lambda_n t)\Bigr|^2dt` .


The constant 3 has furthermore been shown to be optimal in
:cite:`Logan88`
where the reader will find an intensive discussion on this
question. The next lower estimate is also proved there:

 
.. admonition:: Theorem
   :class: thm-tme-emt

   Let :math:`\lambda_1,\cdots,\lambda_N` be :math:`N` be real numbers, and suppose
   that :math:`a_n\ge 0` for all :math:`n`. Then
   :math:`\displaystyle    \int_{-T}^T\Bigl|\sum_{1\le n\le N}a_n e(\lambda_n t)\Bigr|^2dt  \ge  T \sum_{n\le N}a_n^2.` 




We follow the idea of Corollary 3 of
:cite:`Montgomery-Vaughan74`
but rely on
:cite:`Preissmann84` to get the following.

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

   Let :math:`(a_n)_{n\ge1}` be a series of complex numbers that are such that
   :math:`\sum_n n|a_n|^2 < \infty` and :math:`\sum_n |a_n| < \infty`. We have, for :math:`T\ge0`,
   :math:`\displaystyle      \int_0^T\Bigl|    \sum_{n\ge1} a_{n}n^{it}    \Bigr|^{2}dt =     \sum_{n\le N}|a_n|^2 \bigl(T+\mathcal{O}^*(2\pi c_0(n+1))\bigr),` 
   where :math:`c_0=\sqrt{1+\frac23\sqrt{\frac{6}{5}}}`. Moreover, when :math:`a_n` is
   real-valued, the constant :math:`2\pi c_0` may be reduced to :math:`\pi c_0`.


This is Lemma 6.2 from :cite:`Ramare13d`.


 
Corollary 6.3 and 6.4 of
:cite:`Ramare13d`
contain explicit versions of a Theorem of
:cite:`Gallagher70`


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

   Let :math:`(a_n)_{n\ge1}` be a series of complex numbers that are such that
   :math:`\sum_n n|a_n|^2 < \infty` and :math:`\sum_n |a_n| < \infty`. We have, for :math:`T\ge0`,
   :math:`\displaystyle  \sum_{q\le Q}\frac{q}{\varphi(q)}  \sum_{\substack{\chi\mod q,\\ \chi\text{ primitive}}}  \int_{-T}^T    \biggl|\sum_{n}a_n \chi(n)n^{it}\biggr|^2dt    \le    7    \sum_{n}|a_n|^2( n+ Q^2\max(T, 3) ).` 




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

   Let :math:`(a_n)_{n\ge1}` be a series of complex numbers that are such that
   :math:`\sum_n n|a_n|^2 < \infty` and :math:`\sum_n |a_n| < \infty`. We have, for :math:`T\ge0`,
   :math:`\displaystyle  \sum_{q\le Q}\frac{q}{\varphi(q)}  \sum_{\substack{\chi\mod q,\\ \chi\text{ primitive}}}  \int_{-T}^T    \biggl|\sum_{n}a_n \chi(n)n^{it}\biggr|^2dt    \le    \sum_{n}|a_n|^2( 43n+ \tfrac{33}{8} Q^2\max(T, 70) ).`