Large sieve inequality with characters to square moduli. (English) Zbl 1060.11055

Zhao, Liangyi

Acta Arith. 112, No. 3, 297-308 (2004).

In this carefully written paper, the author mainly establishes that \(\sum_{q\leq Q}\sum_{a(q)}| S(a/q^2)| ^2\), where \(S(\alpha)=\sum_{n\leq N}a_ne(n\alpha)\) and \(a(q)\) means that \(a\) ranges through the invertible residue classes modulo \(q\), is not more than \(C_\varepsilon \sum_n| a_n| ^2(Q^3+N^{1+\varepsilon}\sqrt{Q}+N^{\frac12+\varepsilon}Q)\log (2Q)\) for every \(\varepsilon>0\) and where \(C_\varepsilon\) is a positive constant depending on \(\varepsilon\). Corresponding (though weaker) results are given for higher powers as well.
The proof goes first by duality and then by counting the number \(a/q^2\) with \(a\) prime to \(q\) that are close to each other, as in the double large sieve inequality. This counting is done via exponential sums techniques. The paper ends with some numerical evidence pertaining to a conjecture for this number: if the distance is less than \(1/Q^3\) then this number of points is conjectured to be at most \(C_\varepsilon Q^\varepsilon\) for every \(\varepsilon>0\).
This paper is pleasant to read.

Reviewer: Olivier Ramaré (Lille)

MSC:

11N35	Sieves
11L15	Weyl sums
11L40	Estimates on character sums

Keywords:

large sieve inequality