arXiv subjects

Zhishan Yang

Publications and source records attributed to Zhishan Yang.

4 recordsLinked to original sources

arXiv2026

Higher moments for symmetric powers of modular forms

Let $f$ be a cuspidal eigenform of weight $k$ on $\SL_2(\BZ)$ and let $\lambda_{\Sym^d f}(n)$ be the normalized Fourier coefficients of its $d$-th symmetric power lift. This paper establishes asymptotic formulas for the moments $\sum_{n\leq x}\lambda^l_{\Sym^d f}(n)$ for all positive integers $d$ and $l$. We also prove an asymptotic formula for the corresponding sum over the values of any positive definite binary quadratic form $Q$. Our results generalize and improve upon previous work, which was limited to small values of $d$ or $l$. The proofs rely on the decomposition of $\ell$-adic Galois representations and the analytic properties of the associated $L$-functions.

math.NT↗

arXiv2021

On some sums involving the integral part function

Denote by $τ$ k (n), $ω$(n) and $μ$ 2 (n) the number of representations of n as product of k natural numbers, the number of distinct prime factors of n and the characteristic function of the square-free integers, respectively. Let [t] be the integral part of real number t. For f = $ω$, 2 $ω$ , $μ$ 2 , $τ$ k , we prove that n x f x n = x d 1 f (d) d(d + 1) + O $ε$ (x $θ$ f +$ε$) for x $\rightarrow$ $\infty$, where $θ$ $ω$ = 53 110 , $θ$ 2 $ω$ = 9 19 , $θ$ $μ$2 = 2 5 , $θ$ $τ$ k = 5k--1 10k--1 and $ε$ > 0 is an arbitrarily small positive number. These improve the corresponding results of Bordell{è}s.

math.NT↗

arXiv2021

A variant of the prime number theorem

Let $Λ(n)$ be the von Mangoldt function, and let $[t]$ be the integral part of real number $t$. In this note, we prove that for any $\varepsilon>0$ the asymptotic formula $$ \sum_{n\le x} Λ\Big(\Big[\frac{x}{n}\Big]\Big) = x\sum_{d\ge 1} \frac{Λ(d)}{d(d+1)} + O_{\varepsilon}\big(x^{9/19+\varepsilon}\big) \qquad (x\to\infty)$$ holds. This improves a recent result of Bordellès, which requires $\frac{97}{203}$ in place of $\frac{9}{19}$.

math.NT↗

arXiv2015

The ideal counting function in cubic fields

For a cubic algebraic extension $K$ of $\mathbb{Q}$, the behavior of the ideal counting function is considered in this paper. Let $a_{K}(n)$ be the number of integral ideals of the field $K$ with norm $n$. An asymptotic formula is given for the sum $$ \sum\limits_{n_{1}^2+n_{2}^2\leq x}a_{K}(n_{1}^2+n_{2}^2). $$

math.NT↗