PreprintCC BY-NC-SA 4.0arXiv
The large sieve through Parseval, Large sifted sets are regular
Abstract
We modify the approach to the arithmetical form of the large sieve by relying on the Parseval identity rather than on an approximate Bessel inequality and as a consequence, we prove that sifted sets are either small or the attached Fourier polynomial is very regular at the origin. We also discuss the optimality of this approach and shall show in passing that N∫−δδ|S(β)|2dβ≥(1−2/√(δN))|S(0)|2 for any trigonometric polynomial of length~N.
Related
- The weighted large sieve through Parseval
- Optimal large sieve for fixed order characters
- Luo's Spectral Large Sieve Inequality on Short Intervals
- Optimization of Quadratic Sieve Algorithm Implementation for Large Integer Factorization
- The large sieve inequality and additive decompositions of sums of squares
BibTeX
@misc{ramare2026large,
title = {{The large sieve through Parseval, Large sifted sets are regular}},
author = {Olivier Ramaré},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25879},
eprint = {2609.25879},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25879},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - The large sieve through Parseval, Large sifted sets are regular
AU - Olivier Ramaré
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25879
UR - https://arxiv.org/abs/2609.25879
AB - We modify the approach to the arithmetical form of the large sieve by relying on the Parseval identity rather than on an approximate Bessel inequality and as a consequence, we prove that sifted sets are either small or the attached Fourier polynomial is very regular at the origin. We also discuss the optimality of this approach and shall show in passing that N∫_−δ^δ|S(β)|^2dβ≥(1−2/√(δN))|S(0)|^2 for any trigonometric polynomial of length~N.
ER -