Large value estimates for Dirichlet polynomials and trilinear sums over short intervals

Source abstract

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 $θ> \frac{3}{5}$. In this paper, we use their method to improve the Baker--Harman--Pintz estimate for $\frac{1}{2} < θ< \frac{4}{7}$ and get new results for all $\frac{1}{2} < θ< 1$. Our result yields improvements on many problems, including the work of Li on primes in short intervals of length $x^{0.52}$.