On the exponential sum over squarefree integers

Abstract

Let μ be the Möbius function and e(t)=e^2πit. We prove that if N≥2, α∈ℝ, (a,q)=1, and |α−a/q|≤ q⁻², then |∑_{n≤ N}μ²(n)e(αn)|≪(N/q+q)(log 2N)⁵, with an absolute implied constant, and we deduce the corresponding estimate on the minor arcs of the Hardy--Littlewood dissection throughout the range Q≤ N^1/2. The estimates of Schlage-Puchta [SP] and of Tolev [T] have the same dependence on q and Q but carry a factor N^ε. The proof uses Heath-Brown's square sieve with sieving primes confined to an interval (P,2P], where P may be as small as a multiple of log N; a finite Fejér majorant in place of a truncated Fourier series; and, after completion of the character sums, a count of representations that exploits the restriction on the primes in place of the divisor function.

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BibTeX RIS

BibTeX

@misc{robles2026exponential,
  title = {{On the exponential sum over squarefree integers}},
  author = {Nicolas Robles and Alexandru Zaharescu and Dirk Zeindler},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.03961},
  eprint = {2609.03961},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03961},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - On the exponential sum over squarefree integers
AU  - Nicolas Robles
AU  - Alexandru Zaharescu
AU  - Dirk Zeindler
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.03961
UR  - https://arxiv.org/abs/2609.03961
AB  - Let μ be the Möbius function and e(t)=e^2πit. We prove that if N≥2, α∈ℝ, (a,q)=1, and |α−a/q|≤ q^−2, then |∑_n≤ Nμ^2(n)e(αn)|≪(N/q+q)(log 2N)^5, with an absolute implied constant, and we deduce the corresponding estimate on the minor arcs of the Hardy--Littlewood dissection throughout the range Q≤ N^1/2. The estimates of Schlage-Puchta [SP] and of Tolev [T] have the same dependence on q and Q but carry a factor N^ε. The proof uses Heath-Brown's square sieve with sieving primes confined to an interval (P,2P], where P may be as small as a multiple of log N; a finite Fejér majorant in place of a truncated Fourier series; and, after completion of the character sums, a count of representations that exploits the restriction on the primes in place of the divisor function.
ER  -

Abstract

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