arXiv subjects

Tinghao Huang

Publications and source records attributed to Tinghao Huang.

5 recordsLinked to original sources

arXiv2026

First Moment of derivatives of $L$-functions in a nonlinear family

In this paper, we prove an asymptotic formula for the moment of the first order derivative of modular $L$-functions at the center of the critical strip, weighted by generalized divisor functions formed by primitive quadratic characters. Such moments were previously studied by Munshi, which naturally arise in the study of elliptic fibration. To the best of our knowledge, such asymptotic formulae have only been proven in the setting of higher-order derivatives, or under the specialization to dihedral forms.

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arXiv2026

On Ramanujan Primes for Hecke-Maass Cusp Forms

For a primitive Hecke-Maass cusp form $\phi$ of level $N$ with the $n$-th Hecke eigenvalue $\lambda_{\phi}(n)$ and a prime number $p\nmid N$, the celebrated Ramanujan conjecture at $p$ asserts the following sharp upper bound: \[ |\lambda_{\phi}(p)| \leq 2. \] In this work, we determine an upper bound for the least prime $p$ at which the Ramanujan conjecture holds for two or three distinct primitive Hecke-Maass cusp forms simultaneously. Moreover, given a set of distinct primitive Hecke-Maass cusp forms $\{\phi_i\}$, we also provide a lower bound for the lower natural density of the set of primes at which the Ramanujan conjecture holds for at least one of the $\phi_i$'s.

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arXiv2025

Averaging quadratically twisted $L$-values and their derivatives

In this paper, we unconditionally establish an asymptotic formula for the product of the quadratically twisted central $L$-value associated to a holomorphic cusp form $f$, and the quadratically twisted central $L$-derivative to a distinct holomorphic cusp form $g$. This result may be viewed as an extension of \cite{Li-MR4768632}, \cite{Kumar.etc-MR4765788} and \cite{zhou2025momentderivativesquadratictwists}.

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arXiv2023

Counting Divisors in the Outputs of a Binary Quadratic Form

For a fixed natural number $h$, we prove meromorphic continuation of the two-variable Dirichlet series $\sum_m r_2(m) σ_w(m + h) (m + h)^{-s + w}$ to $\mathbb{C}^2$ and use this to obtain asymptotics for $\sum_{m^2 + n^2 \leq X} σ_w(m^2 + n^2 + h)$. We approach this continuation through spectral theory. Our results are comparable to earlier work of Bykovskii, who used different methods to study the sums $\sum_{n^2 \leq X} σ_w(n^2 + h)$.

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arXiv2021

Spherical Heron triangles and elliptic curves

We define spherical Heron triangles (spherical triangles with "rational" side-lengths and angles) and parametrize them via rational points of certain families of elliptic curves. We show that the congruent number problem has infinitely many solutions for most areas in the spherical setting and we find a spherical Heron triangle with rational medians. We also explore the question of spherical triangles with a single rational median or a single a rational area bisector (median splitting the triangle in half), and discuss various problems involving isosceles spherical triangles.

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