arXiv subjects

Vorrapan Chandee

Publications and source records attributed to Vorrapan Chandee.

At least 19 recordsLinked to original sources

arXiv2026

One-level densities of large even and odd orthogonal families of automorphic L-functions

We prove one-level density results for L-functions attached to primitive forms of level q, averaged over square-free q, conditional on the Generalized Riemann Hypothesis (GRH). We treat the even and odd orthogonal families separately and extend the support of the Fourier transform of the test function to (-3,3). This extended support yields the strongest known non-vanishing results for these families of L-functions and their derivatives at the central point, conditional on GRH.

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arXiv2025

The $n^{th}$ centered moments of a large orthogonal family of automorphic $L$-functions

We obtain the $n$th centered moments of one level densities of a large orthogonal family of $L$-functions associated with holomorphic Hecke newforms of level $q$, averaged over $q\sim Q$. We verify the Katz-Sarnak conjecture for these statistics, in the range where the sum of the supports of the Fourier transforms of test functions lies in $(-4, 4)$. In so doing, we need to understand certain phantom oversized terms, which allow us to extract the right off-diagonal contributions. We further need to resolve the combinatorial problem that arises when matching our main terms with random matrix predictions.

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arXiv2024

The sixth moment of Dirichlet L-functions at the central point

In 1970, Huxley obtained a sharp upper bound for the sixth moment of Dirichlet $L$-functions at the central point, averaged over primitive characters $χ$ modulo $q$ and all moduli $q \leq Q$. In 2007, as an application of their ``asymptotic large sieve'', Conrey, Iwaniec and Soundararajan showed that when an additional short $t$-averaging is introduced into the problem, an asymptotic can be obtained. In this paper we show that this extraneous averaging can be removed, and we thus obtain an asymptotic for the original moment problem considered by Huxley. The main new difficulty in our work is the appearance of certain challenging ``unbalanced'' sums that arise as soon as the $t$-aspect averaging is removed.

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arXiv2024

Low-lying zeros of a large orthogonal family of automorphic $L$-functions

We study a new orthogonal family of $L$-functions associated with holomorphic Hecke newforms of level $q$, averaged over $q \asymp Q$. To illustrate our methods, we prove a one level density result for this family with the support of the Fourier transform of the test function being extended to be inside $(-4, 4)$. The main techniques developed in this paper will be useful in developing further results for this family, including estimates for high moments, information on the vertical distribution of zeros, as well as critical line theorems.

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arXiv2023

The eighth moment of Dirichlet L-functions II

We prove an asymptotic formula for the eighth moment of Dirichlet $L$-functions averaged over primitive characters $χ$ modulo $q$, over all moduli $q\leq Q$ and with a short average on the critical line. Previously the same result was shown conditionally on the Generalized Riemann Hypothesis by the first two authors.

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arXiv2022

On Benford's Law for multiplicative functions

We provide a criterion to determine whether a real multiplicative function is a strong Benford sequence. The criterion implies that the $k$-divisor functions, where $k \neq 10^j$, and Hecke eigenvalues of newforms, such as Ramanujan tau function, are strong Benford. Moreover, we deduce from the criterion that the collection of multiplicative functions which are not strong Benford forms a group under pointwise multiplication. In contrast to earlier work, our approach is based on Halász's Theorem.

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arXiv2022

On Montgomery's pair correlation conjecture: a tale of three integrals

We study three integrals related to the celebrated pair correlation conjecture of H. L. Montgomery. The first is the integral of Montgomery's function $F(α, T)$ in bounded intervals, the second is an integral introduced by Selberg related to estimating the variance of primes in short intervals, and the last is the second moment of the logarithmic derivative of the Riemann zeta-function near the critical line. The conjectured asymptotic for any of these three integrals is equivalent to Montgomery's pair correlation conjecture. Assuming the Riemann hypothesis, we substantially improve the known upper and lower bounds for these integrals by introducing new connections to certain extremal problems in Fourier analysis. In an appendix, we study the intriguing problem of establishing the sharp form of an embedding between two Hilbert spaces of entire functions naturally connected to Montgomery's pair correlation conjecture.

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arXiv2021

The second moment of $GL(4) \times GL(2)$ $L$-functions at special points

In this paper, we obtain upper bounds for the second moment of $L(u_j \times ϕ, \frac{1}{2} + it_j)$, where $ϕ$ is a Hecke Maass form for $SL(4, \mathbb Z)$, and $u_j$ is taken from an orthonormal basis of Hecke-Maass forms on $SL(2, \mathbb{Z})$ with eigenvalue $1/4 + t_j^2$. The bounds are consistent with the Lindelöf hypothesis. Previously these types of upper bounds are available for only $GL(n) \times GL(2)$, where $n \leq 3.$

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arXiv2020

$n$-level density of the low-lying zeros of primitive Dirichlet $L$-functions

Katz and Sarnak conjectured that the statistics of low-lying zeros of various family of $L$-functions matched with the scaling limit of eigenvalues from the random matrix theory. In this paper we confirm this statistic for a family of primitive Dirichlet $L$-functions matches up with corresponding statistic in the random unitary ensemble, in a range that includes the off-diagonal contribution. To estimate the $n$-level density of zeros of the $L$-functions, we use the asymptotic large sieve method developed by Conrey, Iwaniec and Soundararajan. For the random matrix side, a formula from Conrey and Snaith allows us to solve the matchup problem.

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arXiv2017

The eighth moment of the family of $Γ_1(q)$-automorphic $L$-functions

We prove a Lindelöf on average bound for the eighth moment of a family of $L$-functions attached to automorphic forms on $GL(2)$, the first time this has been accomplished. Previously, such a bound had been proven for the sixth moment for our family by Djanković and for a similar family by Young. Our proof rests on a new approach which overcomes the lack of perfect orthogonality in the family initially observed by Iwaniec and Xiaoqing Li.

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arXiv2017

The sixth moment of automorphic $L$-functions

In this paper, we consider the $L$-functions $L(s, f)$ where $f$ is an eigenform for the congruence subgroup $Γ_1(q)$. We prove an asymptotic formula for the sixth moment of this family of automorphic $L$-functions.

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arXiv2015

Group structures of elliptic curves over finite fields

It is well-known that if $E$ is an elliptic curve over the finite field $\mathbb{F}_p$, then $E(\mathbb{F}_p)\simeq\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/mk\mathbb{Z}$ for some positive integers $m, k$. Let $S(M,K)$ denote the set of pairs $(m,k)$ with $m\le M$ and $k\le K$ such that there exists an elliptic curve over some prime finite field whose group of points is isomorphic to $\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/mk\mathbb{Z}$. Banks, Pappalardi and Shparlinski recently conjectured that if $K\le (\log M)^{2-ε}$, then a density zero proportion of the groups in question actually arise as the group of points on some elliptic curve over some prime finite field. On the other hand, if $K\ge (\log M)^{2+ε}$, they conjectured that a density one proportion of the groups in question arise as the group of points on some elliptic curve over some prime finite field. We prove that the first part of their conjecture holds in the full range $K\le (\log M)^{2-ε}$, and we prove that the second part of their conjecture holds in the limited range $K\ge M^{4+ε}$. In the wider range $K\ge M^2$, we show that a positive density of the groups in question actually occur.

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arXiv2015

The frequency of elliptic curve groups over prime finite fields

Letting $p$ vary over all primes and $E$ vary over all elliptic curves over the finite field $\mathbb{F}_p$, we study the frequency to which a given group $G$ arises as a group of points $E(\mathbb{F}_p)$. It is well-known that the only permissible groups are of the form $G_{m,k}:=\mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/mk\mathbb{Z}$. Given such a candidate group, we let $M(G_{m,k})$ be the frequency to which the group $G_{m,k}$ arises in this way. Previously, the second and fourth named authors determined an asymptotic formula for $M(G_{m,k})$ assuming a conjecture about primes in short arithmetic progressions. In this paper, we prove several unconditional bounds for $M(G_{m,k})$, pointwise and on average. In particular, we show that $M(G_{m,k})$ is bounded above by a constant multiple of the expected quantity when $m\le k^A$ and that the conjectured asymptotic for $M(G_{m,k})$ holds for almost all groups $G_{m,k}$ when $m\le k^{1/4-ε}$. We also apply our methods to study the frequency to which a given integer $N$ arises as the group order $\#E(\mathbb{F}_p)$.

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arXiv2015

A note on the zeros of zeta and $L$-functions

Let $πS(t)$ denote the argument of the Riemann zeta-function at the point $s=\tfrac12+it$. Assuming the Riemann hypothesis, we give a new and simple proof of the sharpest known bound for $S(t)$. We discuss a generalization of this bound for a large class of $L$-functions including those which arise from cuspidal automorphic representations of GL($m$) over a number field. We also prove a number of related results including bounding the order of vanishing of an $L$-function at the central point and bounding the height of the lowest zero of an $L$-function.

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arXiv2015

Trilinear forms with Kloosterman fractions

We give new bounds for $\sum_{a, m ,n}α_{m}β_nν_a {\textrm e}\left(\frac{a\overline m}{n}\right)$ where $α_{m}$, $β_n$ and $ν_a$ are arbitrary coefficients, improving upon a result of Duke, Friedlander and Iwaniec [DFI97]. We also apply these bounds to problems on representations by determinant equations and on the equidistribution of solutions to linear equations.

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arXiv2014

The mean square of the product of $ζ(s)$ with Dirichlet polynomials

Improving earlier work of Balasubramanian, Conrey and Heath-Brown, we obtain an asymptotic formula for the mean-square of the Riemann zeta-function times an arbitrary Dirichlet polynomial of length $T^{1/2 + δ}$, with $δ= 0.01515....$ As an application we obtain an upper bound of the correct order of magnitude for the third moment of the Riemann zeta-function. We also refine previous work of Deshouillers and Iwaniec, obtaining asymptotic estimates in place of bounds. Using the work of Watt, we compute the mean-square of the Riemann zeta-function times a Dirichlet polynomial of length going up to $T^{3/4}$ provided that the Dirichlet polynomial assumes a special shape. Finally, we exhibit a conjectural estimate for trilinear sums of Kloosterman fractions which implies the Lindelof Hypothesis.

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arXiv2014

Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function

Montgomery's pair correlation conjecture predicts the asymptotic behavior of the function $N(T,β)$ defined to be the number of pairs $γ$ and $γ'$ of ordinates of nontrivial zeros of the Riemann zeta-function satisfying $0<γ,γ'\leq T$ and $0 < γ'-γ\leq 2πβ/\log T$ as $T\to \infty$. In this paper, assuming the Riemann hypothesis, we prove upper and lower bounds for $N(T,β)$, for all $β>0$, using Montgomery's formula and some extremal functions of exponential type. These functions are optimal in the sense that they majorize and minorize the characteristic function of the interval $[-β, β]$ in a way to minimize the $L^1\big(\mathbb{R}, \big\{1 - \big(\frac{\sin πx}{πx}\big)^2 \big\}\,dx\big)$-error. We give a complete solution for this extremal problem using the framework of reproducing kernel Hilbert spaces of entire functions. This extends previous work by P. X. Gallagher in 1985, where the case $β\in \frac12 \mathbb{N}$ was considered using non-extremal majorants and minorants.

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arXiv2014

The eighth moment of Dirichlet $L$-functions

We prove an asymptotic for the eighth moment of Dirichlet $L$-functions averaged over primitive characters $χ$ modulo $q$, over all moduli $q\leq Q$ and with a short average on the critical line, conditionally on GRH. We derive the analogous result for the fourth moment of Dirichlet twists of GL(2) L-functions. Our results match the moment conjectures in the literature; in particular, the constant 24024 appears as a factor in the leading order term of the eighth moment.

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