PreprintCC BY 4.0arXiv
A folklore conjecture on multiplicative exponential sums
Abstract
We prove a folklore conjecture on exponential sums with multiplicative coefficients: if f is multiplicative with |f(n)|≤1, (a,q)=1, |α−a/q|≤ q−2 and q≤ N/(log N)2, then ∑n≤ Nf(n)e(nα)≪N/(log N)+N/√q. Complete multiplicativity is not required. We also prove two related conjectured bounds: the refinement in which q is replaced by q(1+N|α−a/q|), when this quantity is at most N/(log N)2, and the corresponding bound for the sum with logarithmic weights. Finally, we show that the bound can fail for multiplicative f satisfying only |f(p)|≤ A at primes and ∑n≤ x|f(n)|2≤ A2x, for every fixed A>1.
Related
- Logarithmic exponential sums with multiplicative coefficients
- On exponential sums with multiplicative coefficients
- Effective estimates for exponential sums with multiplicative coefficients
- Low moments of automorphic random multiplicative function sums
- A sharp almost sure upper bound for partial sums of random multiplicative functions
BibTeX
@misc{robles2026folklore,
title = {{A folklore conjecture on multiplicative exponential sums}},
author = {Nicolas Robles},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19464},
eprint = {2610.19464},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19464},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A folklore conjecture on multiplicative exponential sums
AU - Nicolas Robles
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19464
UR - https://arxiv.org/abs/2610.19464
AB - We prove a folklore conjecture on exponential sums with multiplicative coefficients: if f is multiplicative with |f(n)|≤1, (a,q)=1, |α−a/q|≤ q^−2 and q≤ N/(log N)^2, then ∑_n≤ Nf(n)e(nα)≪N/(log N)+N/√q. Complete multiplicativity is not required. We also prove two related conjectured bounds: the refinement in which q is replaced by q(1+N|α−a/q|), when this quantity is at most N/(log N)^2, and the corresponding bound for the sum with logarithmic weights. Finally, we show that the bound can fail for multiplicative f satisfying only |f(p)|≤ A at primes and ∑_n≤ x|f(n)|^2≤ A^2x, for every fixed A>1.
ER -