Searcharxiv⌕ Search

arXiv subjects

Peng-Jie Wong

Publications and source records attributed to Peng-Jie Wong.

At least 19 recordsLinked to original sources
arXiv2026

Counting zeros of Artin $L$-functions

In this article, assuming Artin's (holomorphy) conjecture, we establish an explicit asymptotic formula for the number of non-trivial zeros, up to any given height $T\geq 1$, of Artin $L$-functions. As a consequence, our result yields an unconditional explicit zero-counting formula for Hecke $L$-functions over any number field. In addition, our result improves the recent work of Amberger on Dedekind and Riemann zeta functions and the previous work of Bennett-Martin-O'Bryant-Rechnitzer on Dirichlet $L$-functions for sufficiently large $T$.

math.NT↗
arXiv2026

On rates of convergence in central limit theorems of Selberg and Bourgade

Based on the recent works of Radziwill-Soundararajan and Roberts, we establish a rate of convergence in Bourgade's central limit theorem for shifted Dirichlet $L$-functions. Our results also indicate that the dependence structure in the components of a random vector could have a dramatic impact on the rate of convergence in such a multivariate central limit theorem.

math.NT↗
arXiv2025

Towards Keating-Snaith's conjecture for cubic Hecke $L$-functions over the Eisenstein field

A famous conjecture of Keating and Snaith asserts that central values of $L$-functions in a given family admit a log-normal distribution with a prescribed mean and variance depending on the symmetry type of the family. Based on a recent work of Radziwill and Soundararajan, we obtain a conditional lower bound towards Keating-Snaith's conjecture for a "thin" family of cubic Hecke $L$-functions over the Eisenstein field. A key new input is certain twisted estimates of the 1-level density of zeros of cubic Hecke $L$-functions, extending the previous work of David and Güloğlu, under the Generalised Riemann Hypothesis.

math.NT↗
arXiv2025

Square-free orders for CM elliptic curves modulo $p$ in short intervals

Let $E$ be a CM elliptic curve over $\Bbb{Q}$. We refine the work of Cojocaru on the asymptotic formulae for the number of primes $p\le x$ for which the reduction modulo $p$ of $E$ is of square-free order. Also, we derive an unconditional short interval variant for the asymptotics. Compared to the estimate derived from the generalised Riemann hypothesis, the presented result is valid for even shorter intervals. Furthermore, we improve the short interval variant of the cyclicity problem for CM elliptic curves previously obtained by the author.

math.NT↗
arXiv2025

Improved estimates for the argument and zero-counting function of the Riemann zeta-function

In this article, we improve the recent work of Hasanalizade, Shen, and Wong by establishing \[ \left| N (T) - \frac{T}{ 2 π} \log \left( \frac{T}{2πe}\right) \right|\le 0.10076\log T+0.24460\log\log T+8.08344, \] for every $T\ge e$, where $N(T)$ is the number of non-trivial zeros $ρ=β+iγ$, with $0<γ\le T$, of the Riemann zeta-function $ζ(s)$. The main source of improvement comes from implementing new subconvexity bounds for $ζ(σ+it)$ on some $σ_k$-lines inside the critical strip.

math.NT↗
arXiv2024

Low-Lying Zeros of $L$-functions of Adélic Hilbert Modular Forms and their Convolutions

In this article, we study the density conjecture of Katz and Sarnak for $L$-functions of adélic Hilbert modular forms and their convolutions. In particular, under the generalised Riemann hypothesis, we establish several instances supporting the conjecture and extending the works of Iwaniec-Luo-Sarnak and many others. For applications, we obtain an upper bound for the average order of $L$-functions of Hilbert modular forms at $s=\frac{1}{2}$ as well as a positive proportion of non-vanishing of certain Rankin-Selberg $L$-functions.

math.NT↗
arXiv2024

On distributions of $L'$-values and orders of Sha groups in families of quadratic twists

In this article, we aim to establish a prototype result regarding lower bounds of (joint) distributions of central $L'$-values through extending a method of Radziwill and Soundararajan of proving conditional bounds for distributions of central $L$-values (via the one-level density of low-lying zeros of involving $L$-functions). To illustrate this, we give several conditional bounds towards joint distributions of central $L'$-values and orders of Tate-Shafarevich groups in rank-one families of quadratic twists. As an application, we derive a simultaneous non-vanishing result for central $L'$-values in families of quadratic twists of triples of holomorphic modular forms.

math.NT↗
arXiv2024

Cyclicity and exponent of elliptic curves modulo $p$ in arithmetic progressions

In this article, we study the cyclicity problem of elliptic curves $E/\Bbb{Q}$ modulo primes in a given arithmetic progression. We extend the recent work of Akbal and Güloğlu by proving an unconditional asymptotic for such a cyclicity problem over arithmetic progressions for CM elliptic curves $E$, which also presents a generalisation of the previous works of Akbary, Cojocaru, M.R. Murty, V.K. Murty, and Serre. In addition, we refine the conditional estimates of Akbal and Güloğlu, which gives log-power savings (for small moduli) and consequently improves the work of Cojocaru and M.R. Murty. Moreover, we study the average exponent of $E$ modulo primes in a given arithmetic progression and obtain several conditional and unconditional estimates, extending the previous works of Freiberg, Kim, Kurlberg, and Wu.

math.NT↗
arXiv2022

Shifted moments of the Riemann zeta function

In this article, we prove that the Riemann hypothesis implies a conjecture of Chandee on shifted moments of the Riemann zeta function. The proof is based on ideas of Harper concerning sharp upper bounds for the $2k$-th moments of the Riemann zeta function on the critical line.

math.NT↗
arXiv2022

The eighth moment of the Riemann zeta function

In this article, we establish an asymptotic formula for the eighth moment of the Riemann zeta function, assuming the Riemann hypothesis and a quaternary additive divisor conjecture. This builds on the work of the first author on the sixth moment of the Riemann zeta function and work of Conrey-Gonek and Ivić. A key input is a sharp bound for a certain shifted moment of the Riemann zeta function, assuming the Riemann hypothesis.

math.NT↗
arXiv2022

Primes in the Chebotarev density theorem for all number fields

We establish an explicit bound for the least prime occurring in the Chebotarev density theorem without any restriction. Let $L/K$ be any Galois extension of number fields such that $L\not=\mathbb{Q}$, and let $C$ be a conjugacy class in the Galois group of $L/K$. We show that there exists an unramified prime $\mathfrak{p}$ of $K$ such that $σ_{\mathfrak{p}}=C$ and $N \mathfrak{p} \le d_{L}^{B}$ with $B= 310$. This improves the value $B=12\,577$ as proven by Ahn and Kwon. In comparison to previous works on the subject, we modify the weights to detect the least prime, and we use a new version of Turán's power sum method which gives a stronger Deuring-Heilbronn (zero-repulsion) phenomenon. In addition, we refine the analysis of how the location of the potential exceptional zero for $ζ_L(s)$ affects the final result. We also use Fiori's numerical verification for $L$ up to a certain discriminant height. Finally, we provide a lower bound for the number of unramified primes $\mathfrak{p}$ of $K$ such that $σ_{\mathfrak{p}}=C$.

math.NT↗
arXiv2022

Refinements of strong multiplicity one for $\mathrm{GL}(2)$

For distinct unitary cuspidal automorphic representations $π_1$ and $π_2$ for $\mathrm{GL}(2)$ over a number field $F$ and any $α\in\Bbb{R}$, let $\mathcal{S}_α$ be the set of primes $v$ of $F$ for which $λ_{π_1}(v)\neq e^{iα} λ_{π_2}(v)$, where $λ_{π_i}(v)$ is the Fourier coefficient of $π_i$ at $v$. In this article, we show that the lower Dirichlet density of $\mathcal{S}_α$ is at least $\frac{1}{16}$. Moreover, if $π_1$ and $π_2$ are not twist-equivalent, we show that the lower Dirichlet densities of $\mathcal{S}_α$ and $ \cap_α\mathcal{S}_α$ are at least $\frac{2}{13}$ and $\frac{1}{11}$, respectively. Furthermore, for non-twist-equivalent $π_1$ and $π_2$, if each $π_i$ corresponds to a non-CM newform of weight $k_i\ge 2$ and with trivial nebentypus, we obtain various upper bounds for the number of primes $p\le x$ such that $λ_{π_1}(p)^2 = λ_{π_2}(p)^2$. These present refinements of the works of Murty-Pujahari, Murty-Rajan, Ramakrishnan, and Walji.

math.NT↗
arXiv2021

Counting zeros of the Riemann zeta function

In this article, we show that $$ \left| N (T) - \frac{T}{ 2 π} \log \left( \frac{T}{2πe}\right) \right| \le 0.1038 \log T + 0.2573 \log\log T + 9.3675 $$ where $N(T)$ denotes the number of non-trivial zeros $ρ$, with $0<\Im(ρ) \le T$, of the Riemann zeta function. This improves the previous result of Trudgian for sufficiently large $T$. The improvement comes from the use of various subconvexity bounds and ideas from the work of Bennett $et$ $al.$ on counting zeros of Dirichlet $L$-functions.

math.NT↗
arXiv2021

Counting zeros of Dedekind zeta functions

Given a number field $K$ of degree $n_K$ and with absolute discriminant $d_K$, we obtain an explicit bound for the number $N_K(T)$ of non-trivial zeros (counted with multiplicity), with height at most $T$, of the Dedekind zeta function $ζ_K(s)$ of $K$. More precisely, we show that for $T \geq 1$, $$ \Big| N_K (T) - \frac{T}π \log \Big( d_K \Big( \frac{T}{2πe}\Big)^{n_K}\Big)\Big| \le 0.228 (\log d_K + n_K \log T) + 23.108 n_K + 4.520, $$ which improves previous results of Kadiri and Ng, and Trudgian. The improvement is based on ideas from the recent work of Bennett $et$ $al.$ on counting zeros of Dirichlet $L$-functions.

math.NT↗
arXiv2021

Almost all primes satisfy the Atkin-Serre conjecture and are not extremal

Let $f(z)=\sum_{n=1}^{\infty} a_f(n)e^{2πi n z}$ be a non-CM holomorphic cupsidal newform of trivial nebentypus and even integral level $k\geq 2$. Deligne's proof of the Weil conjectures shows that $|a_f(p)|\leq 2p^{\frac{k-1}{2}}$ for all primes $p$. We prove for 100% of primes $p$ that $2p^{\frac{k-1}{2}}\frac{\log\log p}{\sqrt{\log p}}<|a_f(p)|<\lfloor 2p^{\frac{k-1}{2}}\rfloor$. Our proof gives an effective upper bound for the size of the exceptional set. The lower bound shows that the Atkin-Serre conjecture is satisfied for 100% of primes, and the upper bound shows that $|a_f(p)|$ is as large as possible (i.e., $p$ is extremal for $f$) for 0% of primes. Our proofs use the effective form of the Sato-Tate conjecture proved by the second author, which relies on the recent proof of the automorphy of the symmetric powers of $f$ due to Newton and Thorne.

math.NT↗
arXiv2020

On the moments of torsion points modulo primes and their applications

Let $\mathbb{A}[n]$ be the group of $n$-torsion points of a commutative algebraic group $\mathbb{A}$ defined over a number field $F$. For a prime ideal $\mathfrak{p}$, we let $N_{\mathfrak{p}}(\mathbb{A}[n])$ be the number of $\mathbb{F}_\mathfrak{p}$-solutions of the system of polynomial equations defining $\mathbb{A}[n]$ when reduced modulo $\mathfrak{p}$. Here, $\mathbb{F}_{\mathfrak{p}}$ is the residue field at $\mathfrak{p}$. Let $π_F(x)$ denote the number of primes $\mathfrak{p}$ of $F$ whose norm $N(\mathfrak{p})$ do not exceed $x$. We then, for algebraic groups of dimension one, compute the $k$-th moment limit $$M_k(\mathbb{A}/F, n)=\lim_{x\rightarrow \infty} \frac{1}{π_F(x)} \sum_{N(\mathfrak{p}) \leq x} N_{\mathfrak{p}}^k(\mathbb{A}[n])$$ by appealing to the prime number theorem for arithmetic progressions and more generally the Chebotarev density theorem. We further interpret this limit as the number of orbits of the action of the absolute Galois group of $F$on $k$ copies of $\mathbb{A}[n]$ by an application of Burnside's Lemma. These concrete examples suggest a possible approach for determining the number of orbits of a group acting on $k$ copies of a set. We also show that for an algebraic set $Y$ of dimension zero, the corresponding arithmetic function $N_\mathfrak{p}(Y)$, defined on primes $\mathfrak{p}$ of $F$, has an asymptotic limiting distribution.

math.NT↗