PreprintCC BY 4.0arXiv
Logarithmic exponential sums with multiplicative coefficients
Abstract
Let f be a $1$-bounded multiplicative function, and suppose that |α−a/q|≤ 1/(qx), where (a,q)=1 and q≤ x. We prove that |∑n≤ xf(n)/ne( nα)| ≤ (4/√6+o(1))(log x)/√q+O(log q). Here the constant 4/√6 is optimal. This settles two conjectures of Granville and Lamzouri recently. If f is completely multiplicative, the constant improves to $1$. The main new ingredient is a two-prime Ramaré decomposition identity.
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BibTeX
@misc{gou2026logarithmic,
title = {{Logarithmic exponential sums with multiplicative coefficients}},
author = {Haozhe Gou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22081},
eprint = {2610.22081},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22081},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Logarithmic exponential sums with multiplicative coefficients
AU - Haozhe Gou
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22081
UR - https://arxiv.org/abs/2610.22081
AB - Let f be a $1$-bounded multiplicative function, and suppose that |α−a/q|≤ 1/(qx), where (a,q)=1 and q≤ x. We prove that |∑_n≤ xf(n)/ne( nα)| ≤ (4/√6+o(1))(log x)/√q+O(log q). Here the constant 4/√6 is optimal. This settles two conjectures of Granville and Lamzouri recently. If f is completely multiplicative, the constant improves to $1$. The main new ingredient is a two-prime Ramaré decomposition identity.
ER -