PreprintCC BY-NC-SA 4.0arXiv
Large value estimates for Dirichlet polynomials and trilinear sums over short intervals
Abstract
Recently, Matomäki, Pandey, Pliego, Teräväinen and Wang improved the Baker--Harman--Pintz estimate of trilinear sums over short intervals of length xθ, which naturally reduces to estimating mean values of products of three Dirichlet polynomials. Based on recent works of Guth and Maynard and Chen, Gupta and Li on new large value estimates for Dirichlet polynomials, they improved the Baker--Harman--Pintz estimate for θ> 3/5. In this paper, we use their method to improve the Baker--Harman--Pintz estimate for 1/2 < θ< 4/7 and get new results for all 1/2 < θ< 1. Our result yields improvements on many problems, including the work of Li on primes in short intervals of length x0.52.
Related
- Trilinear Kloosterman fractions II: subdyadic intervals and nearly balanced convolutions
- Almost primes in short intervals and averages of long Dirichlet polynomial approximations to (ζ'/ζ)k
- Poisson-Dirichlet statistics for the log-modulus of the Riemann zeta function on expanding intervals near the critical line
- Long intervals of consecutive composite values of polynomials
- Orthogonal polynomials on two or more intervals: the Sprugnoli context
BibTeX
@misc{li2026large,
title = {{Large value estimates for Dirichlet polynomials and trilinear sums over short intervals}},
author = {Runbo Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18923},
eprint = {2610.18923},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18923},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Large value estimates for Dirichlet polynomials and trilinear sums over short intervals
AU - Runbo Li
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.18923
UR - https://arxiv.org/abs/2610.18923
AB - Recently, Matomäki, Pandey, Pliego, Teräväinen and Wang improved the Baker--Harman--Pintz estimate of trilinear sums over short intervals of length x^θ, which naturally reduces to estimating mean values of products of three Dirichlet polynomials. Based on recent works of Guth and Maynard and Chen, Gupta and Li on new large value estimates for Dirichlet polynomials, they improved the Baker--Harman--Pintz estimate for θ> 3/5. In this paper, we use their method to improve the Baker--Harman--Pintz estimate for 1/2 < θ< 4/7 and get new results for all 1/2 < θ< 1. Our result yields improvements on many problems, including the work of Li on primes in short intervals of length x^0.52.
ER -