Tools on Mellin transforms¶

1. Explicit truncated Perron formula¶

Here is Theorem 7.1 of [Ramaré, 2007].

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 \(\displaystyle \sum_{n\le x}a_n =\frac1{2i\pi}\int_{\kappa-iT}^{\kappa+iT}F(z)\frac{x^zdz}z +\mathcal{O}^*\left( \int_{1/T}^{\infty} \sum_{|\log(x/n)|\le u}\frac{|a_n|}{n^\kappa} \frac{2x^\kappa du}{T u^2} \right).\)

See [Ramaré, 2016] for different versions.

2. \(L{}^2\)-means¶

We start with a majorant principle taken for instance from [Montgomery, 1994], 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 \(\displaystyle \int_{-T}^T\Bigl|\sum_{1\le n\le N}a_n e(\lambda_n t)\Bigr|^2dt \le 3 \int_{-T}^T\Bigl|\sum_{1\le n\le N}A_n e(\lambda_n t)\Bigr|^2dt\) .

The constant 3 has furthermore been shown to be optimal in [Logan, 1988] 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 \(\displaystyle \int_{-T}^T\Bigl|\sum_{1\le n\le N}a_n e(\lambda_n t)\Bigr|^2dt \ge T \sum_{n\le N}a_n^2.\)

We follow the idea of Corollary 3 of [Montgomery and Vaughan, 1974] but rely on [Preissmann, 1984] 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 \(T\ge0\), \(\displaystyle \int_0^T\Bigl| \sum_{n\ge1} a_{n}n^{it} \Bigr|^{2}dt = \sum_{n\le N}|a_n|^2 \bigl(T+\mathcal{O}^*(2\pi c_0(n+1))\bigr),\) where \(c_0=\sqrt{1+\frac23\sqrt{\frac{6}{5}}}\). 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].

Corollary 6.3 and 6.4 of [Ramaré, 2016] contain explicit versions of a Theorem of [Gallagher, 1970]

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 \(T\ge0\), \(\displaystyle \sum_{q\le Q}\frac{q}{\varphi(q)} \sum_{\substack{\chi\mod q,\\ \chi\text{ primitive}}} \int_{-T}^T \biggl|\sum_{n}a_n \chi(n)n^{it}\biggr|^2dt \le 7 \sum_{n}|a_n|^2( n+ Q^2\max(T, 3) ).\)

Theorem (2013)