PreprintCC BY 4.0arXiv
On exponential sums with multiplicative coefficients
Abstract
Let f:ℕ→z∈ℂ:|z|≤1 be multiplicative, and suppose that α has a rational approximation a/q with |α−a/q|≤ 1/(qx) and q≤ x/((log x)2(loglog x)3). We prove that |∑n≤ xf(n)e(nα)|≤ (4/√6+o(1))x/√q +O(x/(log x)), whereas complete multiplicativity improves 4/√6 to $1$. Both constants are optimal. In particular, this confirms a folklore conjecture on exponential sums with multiplicative coefficients motivated by the work of Montgomery and Vaughan (1977). We also obtain logarithmically weighted estimates and give counterexamples to related conjectures.
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BibTeX
@misc{gou2026exponential,
title = {{On exponential sums with multiplicative coefficients}},
author = {Haozhe Gou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22345},
eprint = {2610.22345},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22345},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On exponential sums with multiplicative coefficients
AU - Haozhe Gou
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22345
UR - https://arxiv.org/abs/2610.22345
AB - Let f:ℕ→z∈ℂ:|z|≤1 be multiplicative, and suppose that α has a rational approximation a/q with |α−a/q|≤ 1/(qx) and q≤ x/((log x)^2(loglog x)^3). We prove that |∑_n≤ xf(n)e(nα)|≤ (4/√6+o(1))x/√q +O(x/(log x)), whereas complete multiplicativity improves 4/√6 to $1$. Both constants are optimal. In particular, this confirms a folklore conjecture on exponential sums with multiplicative coefficients motivated by the work of Montgomery and Vaughan (1977). We also obtain logarithmically weighted estimates and give counterexamples to related conjectures.
ER -