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Sun-Kai Leung

Publications and source records attributed to Sun-Kai Leung.

13 recordsLinked to original sources
arXiv2026

Low moments of automorphic random multiplicative function sums

We determine the order of the low moments of partial sums of random multiplicative functions associated with Euler products of bounded degree whose unitary local parameters satisfy a prime-square cancellation condition, thereby generalizing Harper's seminal work. The result applies to irreducible compact-group representations of unitary or symplectic type, including the higher-rank Sato--Tate model $SU(d)$ and the odd symmetric-power Sato--Tate model $\operatorname{Sym}^{m}SU(2)$.

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arXiv2026

Low moments of Hecke eigenvalue sums

We show that partial sums of the Sato--Tate random multiplicative functions introduced by Cogdell and Michel exhibit better-than-square-root cancellation. The proof proceeds via a connection to multiplicative chaos, following Harper's seminal work. By a non-trivial adaptation of Harper's derandomization argument for character sums, we also obtain upper bounds for low moments of Hecke eigenvalue sums and of Hecke eigenforms near the cusp $0$; to our knowledge, this is the first appearance of multiplicative chaos in the context of automorphic forms on $\mathrm{GL}(2)$. A novel ingredient is the introduction of Hecke $s$-norms.

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arXiv2026

Value distribution of multiplicative functions along linear fractional sequences

For $a,c\in\mathbb{N}$ and $b,d\in\mathbb{Z}$ such that the (non-empty) set \[ R_{a,b,c,d} :=\left\{\frac{an+b}{\,cn+d\,}: n\in\mathbb{N}\right\} \cap\bigl(\mathbb{Q}_{>0}\setminus\{1\}\bigr) \] is multiplicatively recurrent, we give a complete characterization of the set of limit points of every unimodular multiplicative function $f\in\mathcal{M}$ along $R_{a,b,c,d}.$ We show that the possible limit sets are either the finite subgroups of the unit circle or the entire circle, thereby extending the dichotomy of Klurman--Mangerel.

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arXiv2026

A central limit theorem for prime geodesics on random surfaces of large genus

We show that, for Weil--Petersson random closed hyperbolic surfaces of large genus, the normalized weighted count of prime geodesics with norm in the interval $(X,X+H]$ is asymptotically Gaussian, provided $H \leq X$ and $H/\log X\to\infty$ as $X\to\infty$. In particular, it applies to intervals which are not necessarily short.

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arXiv2026

Joint distribution of primes in multiple short intervals

Assuming the Riemann hypothesis (RH) and the linear independence conjecture (LI), we show that the weighted count of primes in multiple short intervals follows a multivariate Gaussian distribution with weak negative correlations. As an application, we obtain short-interval analogues of many results in the literature on the Shanks--Rényi prime number race, including a sharp phase transition: biased races between primes in short intervals emerge once the number of intervals exceeds an explicit critical threshold. Our result is new even for a single moving interval, particularly under a quantitative formulation of the linear independence conjecture (QLI).

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arXiv2025

Moments of primes in progressions to a large modulus

Assuming a uniform $q$-variant of the prime $k$-tuple conjecture, we compute moments of the number of primes in arithmetic progressions to a large modulus $q$ as the residue classes vary. Consequently, depending on the size of $φ(q)$, the prime count follows either a Gaussian or a Poisson distribution. In particular, the least prime in progressions follows an exponential distribution, with some unexpected discrepancies observed for smooth moduli.

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arXiv2025

Multiplicative recurrence of Möbius transformations

We establish a complete characterization of multiplicative recurrence for images of the positive integers under Möbius transformations, answering a question of Donoso--Le--Moreira--Sun in the negative. As a consequence, we strengthen and extend a Diophantine approximation result of Charamaras--Mountakis--Tsinas, confirming their conjectures.

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arXiv2024

A central limit theorem for coefficients of $L$-functions in short intervals

Assuming the generalized Lindelöf hypothesis (GLH), a weak version of the generalized Ramanujan conjecture and a Rankin--Selberg type partial sum estimate, we establish the normality of the sum of coefficients of a general $L$-function in short intervals of appropriate length. The novelty lies in the degree aspect under GLH. In particular, this generalizes the result of Hughes and Rudnick on lattice point counts in thin annuli.

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arXiv2024

Pseudorandomness of primes at large scales

Assuming a $q$-variant of the prime $k$-tuple conjecture uniformly, we compute mixed moments of the number of primes in disjoint short intervals and progressions, respectively. This involves estimating the mean of singular series along products of lattices, which is of independent interest. As a consequence, we establish the convergence of both sequences of suitably normalized primes to a standard Poisson point process.

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arXiv2024

A note on the standard zero-free region for $L$-functions

In this short note, we establish a standard zero-free region for a general class of $L$-functions for which their logarithms have coefficients with nonnegative real parts, which includes the Rankin--Selberg $L$-functions for unitary cuspidal automorphic representations.

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arXiv2024

Visiting early at prime times

Given an integer $m \geq 2$ and a sufficiently large $q$, we apply a variant of the Maynard--Tao sieve weight to establish the existence of an arithmetic progression with common difference $q$ for which the $m$-th least prime in such progression is $\ll_m q$, which is best possible. As we vary over progressions instead of fixing a particular one, the nature of our result differs from others in the literature. Furthermore, we generalize our result to dynamical systems. The quality of the result depends crucially on the first return time, which we illustrate in the case of Diophantine approximation.

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arXiv2023

Dirichlet law for factorization of integers, polynomials and permutations

Let $k \geq 2$ be an integer. We prove that factorization of integers into $k$ parts follows the Dirichlet distribution $\text{Dir}\left(\frac{1}{k},\ldots,\frac{1}{k}\right)$ by multidimensional contour integration, thereby generalizing the Deshouillers-Dress-Tenenbaum (DDT) arcsine law on divisors where $k=2$. The same holds for factorization of polynomials or permutations. Dirichlet distribution with arbitrary parameters can be modelled similarly.

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