arXiv subjects

Junren Zheng

Publications and source records attributed to Junren Zheng.

4 recordsLinked to original sources

arXiv2026

Mixed incomplete character sums of rational functions with smooth moduli

Let $\chi=\chi_q$ be a primitive character mod $q$ and fix $\Delta>0$. In 1989 Graham and Ringrose gave strong bounds on character sums $\sum_{M<n\leq M+N} \chi(n)$ in intervals of length $N=q^\Delta$ whenever $q$ is squarefree and is sufficiently smooth. Here we show that the smoothness parameter can be taken to be $N^{1-\epsilon}$. We also discuss various generalizations and applications, obtaining best possible results in several aspects.

math.NT↗

arXiv2025

On the Brun--Titchmarsh theorem. II

Denote by $\pi(x;q,a)$ the number of primes $p\leqslant x$ with $p\equiv a\bmod q.$ We prove new upper bounds for $\pi(x;q,a)$ when $q$ is a large prime very close to $\sqrt{x}$, improving upon the classical work of Iwaniec (1982). The proof reduces to bounding a quintilinear sum of Kloosterman sums, to which we introduce a new shifting argument inspired by Vinogradov--Burgess--Karatsuba, going beyond the classical Fourier-analytic approach thanks to a deep algebro-geometric result of Kowalski--Michel--Sawin on sums of products of Kloosterman sums.

math.NT↗

arXiv2024

On the Brun--Titchmarsh theorem. I

The classical Brun--Titchmarsh theorem gives an upper bound, which is of correct order of magnitude in the full range, for the number of primes $p\leqslant x$ satisfying $p\equiv a\bmod q$. We strengthen this inequality for different ranges of $\log q/\log x$, improving upon previous works by Motohashi, Goldfeld, Iwaniec, Friedlander and Iwaniec, and Maynard for general or special moduli. In particular, we are able to beat Iwaniec's barrier $q<x^{9/20-}$, and improve all existing inequalities in the range $x^{9/20}\ll q<x^{1/2-}$ by utilizing bilinear or trilinear structures in the remainder terms of linear sieve. The proof is based on various estimates for character and exponential sums, which we derive by appealing to arithmetic exponent pairs and bilinear forms with algebraic trace functions from $\ell$-adic cohomology, trilinear forms with Kloosterman fractions, and sums of Kloosterman sums from spectral theory of automorphic forms, as well as large value theorem for Dirichlet polynomials.

math.NT↗

arXiv2019

Weak Horseshoe with bounded-gap-hitting times

In this paper, we consider weak horseshoe with bounded-gap-hitting times. For a flow $(M,ϕ)$, it is shown that if the time one map $(M,ϕ_1)$ has weak horseshoe with bounded-gap-hitting times, so is $(M,ϕ_τ)$ for all $τ\neq 0$. In addition, we prove that for an affine homeomorphsim of a compact metric abelian group, positive topological entropy is equivalent to weak horseshoe with bounded-gap-hitting times.

math.DS↗