An overview

Abstract

It is time for us to take some height and look at what we have been doing from farther away. The first approach, through the large sieve inequality, relied on an arithmetical rewriting of

$$\sum\limits_q {\sum\limits_{a\,\bmod *q} {{{\left| {S\left( {a/q} \right)} \right|}^2}\quad \left( {S\left( \alpha \right) = \sum\limits_n {{u_n}e\left( {n\alpha } \right)} } \right)} }$$

. This rewriting did in fact handle the sum W(q) = ∑_{a mod*q} |S(a/q)|² as one single term, and we tried to maximize it in the subsequent analysis. More precisely, whenever (u _n ) vanishes outside of a given compact set, we prove a useful lower bound for this quantity.