Effective estimates for exponential sums with multiplicative coefficients

@misc{robles2026effective,
  title = {{Effective estimates for exponential sums with multiplicative coefficients}},
  author = {Nicolas Robles},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11070},
  eprint = {2609.11070},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11070},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Effective estimates for exponential sums with multiplicative coefficients
AU  - Nicolas Robles
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11070
UR  - https://arxiv.org/abs/2609.11070
AB  - Let f be multiplicative, with |f(p)|≤ A at primes and ∑_n≤ x|f(n)|^2≤ A^2x for every x≥1. If |α−a/q|≤ q^−2, (a,q)=1, and 3≤ R≤ q≤ N/R, we prove ∑_n≤ Nf(n)e(nα) ≪_A N/(log N) +N/√R√(loglog(3R)) with effective implied constants. Montgomery and Vaughan proved this with second term NR^−1/2(log R)^3/2, and, for $1$-bounded functions, Bachman replaced it by NR^−1/2√(log Rloglog R). We remove the factor √(log R) from Bachman's second term while retaining the original coefficient hypotheses of Montgomery and Vaughan. A more precise estimate records the distance from a rational number. The proof combines the Brun-Titchmarsh inequality on short intervals with maximal Fourier estimates derived from the Carleson-Hunt theorem; the local bounds permit arbitrary prime-dependent prefixes. We also prove sharpness of the square-root displacement dependence.
ER  -