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Bingrong Huang

Publications and source records attributed to Bingrong Huang.

32 records · Page 2Linked to original sources
arXiv2020

Quantum variance for dihedral Maass forms

We establish an asymptotic formula for the weighted quantum variance of dihedral Maass forms on $Γ_0(D) \backslash \mathbb H$ in the large eigenvalue limit, for certain fixed $D$. As predicted in the physics literature, the resulting quadratic form is related to the classical variance of the geodesic flow on $Γ_0(D) \backslash \mathbb H$, but also includes factors that are sensitive to underlying arithmetic of the number field $\mathbb Q(\sqrt{D})$.

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arXiv2020

Averages of coefficients of a class of degree 3 L-functions

In this note, we give a detailed proof of an asymptotic for averages of coefficients of a class of degree three $L$-functions which can be factorized as a product of a degree one and a degree two $L$-functions. We emphasize that we can break the $1/2$-barrier in the error term, and we get an explicit exponent.

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arXiv2019

Sup-norm and nodal domains of dihedral Maass forms

In this paper, we improve the sup-norm bound and the lower bound of the number of nodal domains for dihedral Maass forms, which are a distinguished sequence of Laplacian eigenfunctions on an arithmetic hyperbolic surface. More specifically, let $ϕ$ be a dihedral Maass form with spectral parameter $t_ϕ$, then we prove that $\|ϕ\|_\infty \ll t_ϕ^{3/8+\varepsilon} \|ϕ\|_2$, which is an improvement over the bound $t_ϕ^{5/12+\varepsilon} \|ϕ\|_2$ given by Iwaniec and Sarnak. As a consequence, we get a better lower bound for the number of nodal domains intersecting a fixed geodesic segment under the Lindelöf Hypothesis. Unconditionally, we prove that the number of nodal domains grows faster than $t_ϕ^{1/8-\varepsilon}$ for any $\varepsilon>0$ for almost all dihedral Maass forms.

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arXiv2018

Non-vanishing of Maass form L-functions at the critical point

In this paper, we consider the family $\{L_j(s)\}_{j=1}^{\infty}$ of $L$-functions associated to an orthonormal basis $\{u_j\}_{j=1}^{\infty}$ of even Hecke-Maass forms for the modular group $SL(2, Z)$ with eigenvalues $\{λ_j=κ_{j}^{2}+1/4\}_{j=1}^{\infty}$. We prove the following effective non-vanishing result: At least $50 \%$ of the central values $L_j(1/2)$ with $κ_j \leq T$ do not vanish as $T\rightarrow \infty$. Furthermore, we establish effective non-vanishing results in short intervals.

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arXiv2018

Hybrid subconvexity bounds for twisted $L$-functions on $GL(3)$

Let $q$ be a large prime, and $χ$ the quadratic character modulo $q$. Let $ϕ$ be a self-dual Hecke--Maass cusp form for $SL(3,\mathbb{Z})$, and $u_j$ a Hecke--Maass cusp form for $Γ_0(q)\subseteq SL(2,\mathbb{Z})$ with spectral parameter $t_j$. We prove the hybrid subconvexity bounds for the twisted $L$-functions \[ L(1/2,ϕ\times u_j\timesχ)\ll_{ϕ,\varepsilon} (qt_j)^{3/2-θ+\varepsilon},\quad L(1/2+it,ϕ\timesχ)\ll_{ϕ,\varepsilon} (qt)^{3/4-θ/2+\varepsilon}, \] for any $\varepsilon>0$, where $θ=1/23$ is admissible.

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arXiv2018

Quantum variance for Eisenstein Series

In this paper, we prove an asymptotic formula for the quantum variance for Eisenstein series on $\mathrm{PSL}_2(\mathbb{Z})\backslash \mathbb{H}$. The resulting quadratic form is compared with the classical variance and the quantum variance for cusp forms. They coincide after inserting certain subtle arithmetic factors, including the central values of certain L-functions.

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arXiv2018

Prime lattice points in ovals

We study the distribution of lattice points with prime coordinates lying in the dilate of a convex planar domain having smooth boundary, with nowhere vanishing curvature. Counting lattice points weighted by a von Mangoldt function gives an asymptotic formula, with the main term being the area of the dilated domain, and our goal is to study the remainder term. Assuming the Riemann Hypothesis, we give a sharp upper bound, and further assuming that the positive imaginary parts of the zeros of the Riemann zeta functions are linearly independent over the rationals allows us to give a formula for the value distribution function of the properly normalized remainder term.

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arXiv2018

Super-positivity of a family of L-functions in the level aspect

An automorphic self dual L-function has the super-positivity property if all derivatives of the completed L-function at the central point $s=1/2$ are non-negative and all derivatives at a real point $s > 1/2$ are positive. In this paper we prove that at least 12% of L-functions associated to Hecke basis cusp forms of weight $2$ and large prime level $q$ have the super-positivity property. It is also shown that at least 49% of such L-functions have no real zeros on $ \Re(s) > 0$ except possibly at $s = 1/2.$

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arXiv2017

Super-positivity of a family of L-functions

Zhiwei Yun and Wei Zhang introduced the notion of "super-positivity of self dual L-functions" which specifies that all derivatives of the completed L-function (including Gamma factors and power of the conductor) at the central value $s = 1/2$ should be non-negative. They proved that the Riemann hypothesis implies super-positivity for self dual cuspidal automorphic L-functions on $GL(n)$. Super-positivity of the Riemann zeta function was established by Pólya in 1927 and since then many other cases have been found by numerical computation. In this paper we prove, for the first time, that there are infinitely many L-functions associated to modular forms for $SL(2, \mathbb{Z})$ each of which has the super-positivity property. Our proof also establishes that all derivatives of the completed L-function at any real point $σ> 1/2$ must be positive.

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arXiv2016

Sup-norm bounds for Eisenstein series

The paper deals with establishing bounds for Eisenstein series on congruence quotients of the upper half plane, with control of both the spectral parameter and the level. The key observation in this work is that we exploit better the structure of the amplifier by just supporting on primes for the Eisenstein series, which can use both the analytic method as Young did to get a lower bound for the amplifier and the geometric method as Harcos--Templier did to obtain a more efficient treatment for the counting problem.

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arXiv2016

Exponential sums over primes in short intervals and an application to the Waring--Goldbach problem

Let $Λ(n)$ be the von Mangoldt function, $x$ real and $2\leq y \leq x$. This paper improves the estimate on the exponential sum over primes in short intervals \[ S_k(x,y;α) = \sum_{x< n \leq x+y} Λ(n) e\left( n^k α\right) \] when $k\geq 3$ for $α$ in the minor arcs. And then combined with the Hardy--Littlewood circle method, this enables us to investigate the Waring--Goldbach problem of representing a positive integer $n$ as the sum of $s$ $k$th powers of almost equal prime numbers, which improves the results in Wei and Wooley [12].

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arXiv2015

Strong orthogonality between the Mobius function and nonlinear exponential functions in short intervals

Let $μ(n)$ be the Möbius function, $e(z) = \exp(2πiz)$, $x$ real and $2\leq y \leq x$. This paper proves two sequences $(μ(n))$ and $(e(n^k α))$ are strongly orthogonal in short intervals. That is, if $k \geq 3$ being fixed and $y\geq x^{1-1/4+\varepsilon}$, then for any $A>0$, we have \[ \sum_{x< n \leq x+y} μ(n) e\left(n^k α\right) \ll y(\log y)^{-A} \] uniformly for $α\in \mathbb{R}$.

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