The TME-EMT project

Navigation

Back to top

Explicit truncated Perron formula

L${}^2$-means

---

Tools on Mellin transforms

---

Here is Theorem 7.1 of .
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 for different versions. We start with a majorant principle taken for instance from , 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 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 but rely on 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 .
Corollary 6.3 and 6.4 of contain explicit versions of a Theorem of
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 $T\ge0$, $$ \sum_{q\le Q}\frac{q}{\varphi(q)} \sum_{\substack{\chi\mod q,\\ \text{$\chi$ primitive}}} \int_{-T}^T \biggl|\sum_{n}a_n \chi(n)n^{it}\biggr|^2dt \le \sum_{n}|a_n|^2( 43n+ \tfrac{33}{8} Q^2\max(T, 70) ). $$

---