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Yuk-Kam Lau

Publications and source records attributed to Yuk-Kam Lau.

10 recordsLinked to original sources
arXiv2026

Prime detecting quasi-modular forms in higher level

In a previous work, the authors resolved a conjecture about the structure of prime-detecting quasi-modular forms by studying sign changes occurring in quasi-modular cusp forms. In this paper, we extend the considerations to prime-detecting quasi-modular forms of higher level, in particular describing the structure of the space of quasi-modular forms that detect primes in various arithmetic progressions. We also provide an ``analytic'' proof of the level one case.

math.NT↗
arXiv2026

Divisor problems for restricted Fourier coefficients of modular forms

Let $d(n)$ be the number of divisors of $n$. We investigate the average value of $d(a_f(p))^r$ for $r$ a positive integer and $a_f(p)$ the $p$-th Fourier coefficient of a cuspidal eigenform $f$ having integral Fourier coefficients, where $p$ is a prime subject to a constraint on the angle associated with the normalized Fourier coefficient.

math.NT↗
arXiv2025

Equidistribution of Kloosterman sums over function fields

We prove the Sato--Tate distribution of Kloosterman sums over function fields with explicit error terms, when the places vary in arithmetic progressions or short intervals. A joint Sato--Tate distribution of two ``different" exponential sums is also proved.

math.NT↗
arXiv2025

On a conjecture about prime-detecting quasimodular forms

Motivated by weighted partition of $n$ that vanish if and only if $n$ is a prime, Craig, van Ittersum, and Ono conjecture a classification of quasimodular forms which detect primes in the sense that the $n$-th Fourier coefficient vanishes if and only if $n$ is a prime. In this paper, we prove this conjecture by showing that Fourier coefficients of quasimodular cusp forms exhibit infinitely many sign changes.

math.NT↗
arXiv2016

Local Behavior of Arithmetical Functions with Applications to Automorphic L-Functions

We derive a Voronoi-type series approximation for the local weighted mean of an arithmetical function that is associated to Dirichlet series satisfying a functional equation with gamma factors. The series is exploited to study the oscillation frequency with a method of Heath-Brown and Tsang [7]. A by-product is another proof for the well-known result of no element in the Selberg class of degree 0 \textless{} d \textless{} 1. Our major applications include the sign-change problem of the coefficients of automorphic L-functions for GL m , which improves significantly some results of Liu and Wu [14]. The cases of modular forms of half-integral weight and Siegel eigenforms are also considered.

math.NT↗