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Explicit truncated Perron formula

L${}^2$-means

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Tools on Mellin transforms

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Here is Theorem 7.1 of [Ramaré, 2007 †Ramaré, O. 2007
Eigenvalues in the large sieve inequality
Funct. Approximatio, Comment. Math., 37, 7--35.].
Theorem (2007)

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

See [Ramaré, 2016 †Ramaré, O. 2016
Modified truncated Perron formulae
Ann. Blaise Pascal, 23(1), 109--128.] for different versions. We start with a majorant principle taken for instance from [Montgomery, 1994 †Montgomery, H.L. 1994
Ten lectures on the interface between analytic number theory and harmonic analysis
CBMS Regional Conference Series in Mathematics, vol. 84. Published for the Conference Board of the Mathematical Sciences, Washington, DC.], chapter 7, Theorem 3.
Theorem

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

The constant 3 has furthermore been shown to be optimal in [Logan, 1988 †Logan, B. F. 1988
An interference problem for exponentials
Michigan Math. J., 35(3), 369--393.] where the reader will find an intensive discussion on this question. The next lower estimate is also proved there:
Theorem

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

We follow the idea of Corollary 3 of [Montgomery & Vaughan, 1974 †Montgomery, H.L., & Vaughan, R.C. 1974
Hilbert's inequality
J. Lond. Math. Soc., II Ser., 8, 73--82.] but rely on [Preissmann, 1984 †Preissmann, E. 1984
Sur une inégalité de Montgomery et Vaughan
Enseign. Math., 30, 95--113.] to get the following.
Theorem (2013)

Let $(a_n)_{n\ge1}$ be a series of complex numbers that are such that $\sum_n n|a_n|^2 < \infty$ and $\sum_n |a_n| < \infty$. We have, for $T\ge0$, \begin{equation*} \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), \end{equation*} where $c_0=\sqrt{1+\frac23\sqrt{\frac{6}{5}}}$. Moreover, when $a_n$ is real-valued, the constant $2\pi c_0$ may be reduced to $\pi c_0$.

This is Lemma 6.2 from [Ramaré, 2016 †Ramaré, O. 2016
An explicit density estimate for Dirichlet $L$-series
Math. Comp., 85(297), 335--356.].
Corollary 6.3 and 6.4 of [Ramaré, 2016 †Ramaré, O. 2016
An explicit density estimate for Dirichlet $L$-series
Math. Comp., 85(297), 335--356.] contain explicit versions of a Theorem of [Gallagher, 1970 †Gallagher, P.X. 1970
A large sieve density estimate near $\sigma =1$
Invent. Math., 11, 329--339.]
Theorem (2013)

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

Theorem (2013)

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

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