arXiv subjects

Tengyou Zhu

Publications and source records attributed to Tengyou Zhu.

6 recordsLinked to original sources

arXiv2026

Triple divisor type functions in arithmetic progressions to prime power moduli

We study the equidistribution in arithmetic progressions modulo powers of a fixed odd prime of Hecke eigenvalues of noncuspidal $\mathrm{GL}_3$ automorphic forms, with a focus on the convolution coefficients $λ_{1\boxplus f}(n)=(λ_f\star 1)(n)$,attached to a level $1$ holomorphic primitive cusp form $f$. Utilizing techniques introduced by Kowalski--Lin--Michel and Milićević, we prove the exponent $\vartheta=1/2+7/458-\varepsilon$ is admissible.

math.NT↗

arXiv2026

On the exponent of distribution for convolutions of $\operatorname{GL}(2)$ coefficients to smooth moduli

Let $(λ_f(n))_{n\geqslant1}$ be the Hecke eigenvalues of a holomorphic cusp form $f$. We prove that the exponent of distribution of $λ_f*1$ in arithmetic progressions is as large as $\frac{1}{2}+\frac{1}{70}$ when the modulus $q$ is square-free and has only sufficiently small prime factors.

math.NT↗

arXiv2026

Rankin--Selberg coefficients in arithmetic progressions modulo prime powers

Let $\varepsilon>0$ be given. For prime power moduli $q=p^k$ with $k\geq 2$ and $p\neq 3$, and assuming the Ramanujan--Petersson conjecture for $\GL_2$ Maass forms, we prove that the Rankin--Selberg coefficients $\{λ_f(n)^2\}_{n\geq 1}$ have a level of distribution $θ=2/5+3/305-\varepsilon$ in arithmetic progressions $n \equiv a \bmod q$.

math.NT↗

arXiv2025

Subconvexity for $\rm GL_2 \times GL_2$ $L$-functions in the depth aspect

Let $f$ and $g$ be holomorphic or Maass cusp forms for $\rm SL_2(\mathbb{Z})$ and let $χ$ be a primitive Dirichlet character of prime power conductor $q=p^n$. For any given $\varepsilon>0$, we establish the following subconvexity bound \begin{equation*} L(1/2,f\otimes g \otimes χ)\ll_{f,g,\varepsilon}q^{9/10+\varepsilon}. \end{equation*} The proof employs the DFI circle method with standard manipulations, including the conductor-lowering mechanism, Voronoi summation, and Cauchy--Schwarz inequality. The key input is certain estimates on the resulting character sums, obtained using the $p$-adic version of the van der Corput method.

math.NT↗

arXiv2024

The distribution of powers of primes related to the Frobenius problem

Let $1<c<d$ be two relatively prime integers, $g_{c,d}=cd-c-d$ and $\mathbb{P}$ is the set of primes. For any given integer $k \geq 1$, we prove that $$\#\left\{p^k\le g_{c,d}:p\in \mathbb{P}, ~p^k=cx+dy,~x,y\in \mathbb{Z}_{\geqslant0} \right\}\sim \frac{k}{k+1}\frac{g^{1/k}}{\log g} \quad (\text{as}~c\rightarrow\infty),$$ which gives an extension of a recent result of Ding, Zhai and Zhao.

math.NT↗

arXiv2022

Hybrid subconvexity bounds for twists of $\rm GL(3)$ $L$-functions

Let $π$ be a $SL(3,\mathbb Z)$ Hecke-Maass cusp form and $χ$ a primitive Dirichlet character of prime power conductor $\mathfrak{q}=p^k$ with $p$ prime. In this paper we will prove the following subconvexity bound $$ L\left(\frac{1}{2}+it,π\times χ\right)\ll_{π,\varepsilon} p^{3/4}\big(\mathfrak{q}(1+|t|)\big)^{3/4-3/40+\varepsilon}, $$ for any $\varepsilon >0$ and $t \in \mathbb{R}$.

math.NT↗