Discorrelation of the Möbius function with linear phases in short intervals
Abstract
We introduce a new approach to studying correlations between arithmetic functions and linear phases with respect to highly irrational frequencies over short intervals. As an application, we prove that |∑x<n ≤ x+Hμ(n) e(nα)| ≪ H(log x)−1/3+ε whenever x0.58+ε ≤ H≤ x. In the same range of H we also show that |∑x<n ≤ x+H Λ(n) e(nα)| ≪ H(log x)−A for arbitrarily large A>0 unless there exists an integer 1≤ q≪ (log x)OA(1) such that |qα| ≪ x (log x)OA(1)/H2. This breaks the 3/5 barrier in Theorem 1.5 of arXiv:1911.09076v2 and Theorem 2 of [T. Zhan, "On the representation of large odd integer as a sum of three almost equal primes," Acta Mathematica Sinica 7.3 (1991), 259-272]. Moreover, by our method, any improvement in large value estimates for character-twisted Dirichlet polynomials on the critical range would lead to a corresponding improvement in the lower bound for H.
Related
- Higher order uniformity of the primes and cancellation of the Möbius function in shorter intervals
- Möbius functions for pseudo-Levi subgroups in finite general linear and symplectic groups
- A Möbius function computing h-polynomials of nestohedra
- Primes and the Möbius function over quadratic polynomials
- Short sums of the Möbius function over function fields
BibTeX
@misc{pliego2026discorrelation,
title = {{Discorrelation of the Möbius function with linear phases in short intervals}},
author = {Javier Pliego and Mengdi Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11487},
eprint = {2610.11487},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11487},
}
RIS
TY - UNPB
TI - Discorrelation of the Möbius function with linear phases in short intervals
AU - Javier Pliego
AU - Mengdi Wang
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.11487
UR - https://arxiv.org/abs/2610.11487
AB - We introduce a new approach to studying correlations between arithmetic functions and linear phases with respect to highly irrational frequencies over short intervals. As an application, we prove that |∑_x<n ≤ x+Hμ(n) e(nα)| ≪ H(log x)^−1/3+ε whenever x^0.58+ε ≤ H≤ x. In the same range of H we also show that |∑_x<n ≤ x+H Λ(n) e(nα)| ≪ H(log x)^−A for arbitrarily large A>0 unless there exists an integer 1≤ q≪ (log x)^O_A(1) such that |qα| ≪ x (log x)^O_A(1)/H^2. This breaks the 3/5 barrier in Theorem 1.5 of arXiv:1911.09076v2 and Theorem 2 of [T. Zhan, "On the representation of large odd integer as a sum of three almost equal primes," Acta Mathematica Sinica 7.3 (1991), 259-272]. Moreover, by our method, any improvement in large value estimates for character-twisted Dirichlet polynomials on the critical range would lead to a corresponding improvement in the lower bound for H.
ER -