arXiv subjects

Xiumin Ren

Publications and source records attributed to Xiumin Ren.

3 recordsLinked to original sources

arXiv2024

Roth-type Theorem for high-power system in Piatetski-Shapiro primes (II)

We consider the nonlinear system $c_1p_1^d +c_2p_2^d + \dots + c_s p_s^d = 0$ with $c_1, c_2,\dots, c_s\in\mathbb Z$ being nonzero and satisfying $c_1 +c_2 + \dots + c_s = 0$. We show that for $s\ge 2\lfloor \frac{d^2}2\rfloor+1$ and $c\in\left(1, 1+c(d,s)\right)$, if the system has only $K$-trivial solutions in subset $\mathcal{A}$ of Piatetski-Shapiro primes up to $x$ and corresponding to $c$, then $|\mathcal{A}| \ll \frac{x^{\frac1c}}{\log x} $$\left(\log \log \log \log x\right)^{\frac{2-s}{dc}+\varepsilon}$.

math.NT↗

arXiv2022

Averages of exponential twists of the von Mangoldt function

In this paper, we obtain some improved results for the exponential sum $\sum_{x<n\leq 2x}Λ(n)e(αk n^θ)$ with $θ\in(0,5/12),$ where $Λ(n)$ is the von Mangoldt function. Such exponential sums have relations with the so-called quasi-Riemann hypothesis and were considered by Vinogradov \cite{Va} and Murty-Srinivas \cite{Mu}.

math.NT↗

arXiv2014

Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for $GL_m(\mathbb Z)$

Let $f$ be a full-level cusp form for $GL_m(\mathbb Z)$ with Fourier coefficients $A_f(n_1,...,n_{m-1})$. In this paper an asymptotic expansion of Voronoi's summation formula for $f$ is established. As applications of this formula, a smoothly weighted average of $A_f(n,1,...,1)$ against $e(α|n|^β)$ is proved to be rapidly decayed when $0<β<1/m$. When $β=1/m$ and $α$ equals or approaches $\pm mq^{1/m}$ for a positive integer $q$, this smooth average has a main term of the size of $|A_f(1,...,1,q)+A_f(1,...,1,-q)|X^{1/(2m)+1/2}$, which is a manifestation of resonance of oscillation exhibited by the Fourier coefficients $A_f(n,1,...,1)$. Similar estimate is also proved for a sharp-cut sum.

math.NT↗