Farrell Brumley
We specify sufficient conditions for the square modulus of the local parameters of a family of GL(n) cusp forms to be bounded on average. These conditions are global in nature and are at present satisfied for n less than or equal to 4. As an application, we show that Rankin-Selberg L-functions on GL(m) x GL(n), when m and n are less than or equal to 4, satisfy the standard convexity bound.
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We find the average value of the $L^4$-norm of holomorphic Hecke cusp forms of weight within a dyadic interval, up to an error which saves a power of the weight.
We study the asymptotic behaviour of the twisted first moment of central $L$-values associated to cusp forms in weight aspect on average. Our estimate of the error term allows extending the logarithmic length of mollifier $Δ$ up to 2. The best previously known result, due to Iwaniec and Sarnak, was $Δ<1$. The proof is based on a representation formula for the error in terms of Legendre polynomials.
Given a maximal even-integral lattice $\cL$ of signature $(m+, 2-)$ with an odd $m\geq 3$, we consider the holomorphic cusp forms $F$ of weight $l$ on the bounded symmetric domain of type IV of dimension $m$ with respect to the discriminant subgroup of the orthogonal group $O(\cL)$ defined by $\cL$. Under a non-negativity assumption on the central $L$-values, we prove an equidistribution result of Satake parameters in an ensemble constructed from the central values of standard $L$-functions and the square of the Whittaker-Bessel periods.
In this article, we obtain certain estimate for the shifted convolution sum involving the Fourier coefficients of half-integral weight cusp forms.
Let $Γ\subset\mathrm{PSL}_{2}(\mathbb{R})$ be a Fuchsian subgroup of the first kind acting on the upper half-plane $\mathbb{H}$. Consider the $d_{2k}$-dimensional space of cusp forms $\mathcal{S}_{2k}^Γ$ of weight $2k$ for $Γ$, and let $\{f_{1},\ldots,f_{d_{2k}}\}$ be an orthonormal basis of $\mathcal{S}_{2k}^Γ$ with respect to the Petersson inner product. In this paper we will give effective upper and lower bounds for the supremum of the quantity $S_{2k}^Γ(z):=\sum_{j=1}^{d_{2k}}\vert f_{j}(z)\vert^{2}\,\mathrm{Im}(z)^{2k}$ as $z$ ranges through $\mathbb{H}$.
We estimate the mean square of a short exponential sum involving Fourier coefficients of a cusp form with a linear twist, a smooth weight function, and a relatively short averaging interval.
Let $f$ be a holomorphic cusp form of integral weight $k \geq 3$ for $Γ_{0}(N)$ with nebentypus character $ψ$. Generalising work of Kohnen and Raghuram we construct a kernel function for the $L$-function $L(f,χ,s)$ of $f$ twisted by a primitive Dirichlet character $χ$ and use it to show that the average $\sum_{f \in S_{k}(N,ψ)}\frac{L(f,χ,s)}{<f,f>}\bar{a_{f}(1)}$ over an orthogonal basis of $S_{k}(N,ψ)$ does not vanish on certain line segments inside the critical strip if the weight $k$ or the level $N$ is big enough. As another application of the kernel function we prove an averaged version …
In this article, we establish an average behaviour of the normalised Fourier coefficients of the Hecke eigenforms supported at the integers represented by any primitive integral positive definite binary quadratic form of fixed discriminant $D < 0$ when the class number $h(D) = 1$. We also obtain a quantitative result for the number of sign changes of the sequence of the normalised Fourier coefficients $λ_{f}(n)$ of the Hecke eigenforms $f$ where $n$ is represented by any primitive integral positive definite binary quadratic form of fixed discriminant $D < 0$ when the class number $h(D) = 1$ in…
In this article, we derive a sub convexity estimate of Hecke eigen cusp forms associated to certain cocompact arithmetic subgroups of SL(2,R). The main result can be considered as the holomorphic version of the estimate of Hecke eigen Maass forms, derived in a famous paper of Iwaniec and Sarnak. A stronger estimate was derived by Khayutin and Steiner in arXiv:2009.07194. However, techniques used in both the papers are very different.
Let $Γ\subseteq\mathrm{PSL}_{2}(\mathbb{R})$ be a Fuchsian subgroup of the first kind acting on the upper half-plane $\mathbb{H}$. Consider the $d$-dimensional space of cusp forms $\mathcal{S}_{k}^Γ$ of weight $2k$ for $Γ$, and let $\{f_{1},\ldots,f_{d}\}$ be an orthonormal basis of $\mathcal{S}_{k}^Γ$ with respect to the Petersson inner product. In this paper we show that the sup-norm of the quantity $S_{k}^Γ(z):=\sum_{j=1}^{d}| f_{j}(z)|^{2}\,\mathrm{Im}(z)^{2k}$ is bounded as $O_Γ(k)$ in the cocompact setting, and as $O_Γ(k^{3/2})$ in the cofinite case, where the implied constants depend so…
We prove a bound for the Fourier coefficients of a cusp form of integral weight which is not a newform by computing an explicit orthogonal basis for the space of cusp forms of given integral weight and level. In contrast to previous work on special cases of this problem we use elementary methods for the computation of Petersson products and avoid using the Rankin $L$-function.
Let $f$ be a holomorphic cusp form of weight $k$ with respect to $SL_2(\mathbb{Z})$ which is a normalized Hecke eigenform, $L_f(s)$ the $L$-function attached to the form $f$. In this paper, we shall give the relation of the number of zeros of $L_f(s)$ and the derivatives of $L_f(s)$ using Berndt's method, and an estimate of zero-density of the derivatives of $L_f(s)$ based on Littlewood's method.
Let $f$ be a weight $k$ holomorphic cusp form of level one, and let $S_f(n)$ denote the sum of the first $n$ Fourier coefficients of $f$. In analogy with Dirichlet's divisor problem, it is conjectured that $S_f(X) \ll X^{\frac{k-1}{2} + \frac{1}{4} + ε}$. Understanding and bounding $S_f(X)$ has been a very active area of research. The current best bound for individual $S_f(X)$ is $S_f(X) \ll X^{\frac{k-1}{2} + \frac{1}{3}} (\log X)^{-0.1185}$ from Wu. Chandrasekharan and Narasimhan showed that the Classical Conjecture for $S_f(X)$ holds on average over intervals of length $X$. Jutila improved …
A zero-density estimate for $L$-functions associated with $\rm GL(3)$ Hecke--Maass cusp forms
研究通过建立 $\rm SL(3,\mathbb{Z})$ Hecke–Maass 尖点形式对应 $L$-函数的扭曲二阶矩渐近公式，得到其谱方面的加权零点密度估计。
In this paper, we study the average behaviour of the coefficients of triple product L-functions and some related L-functions corresponding to normalized primitive holomorphic cusp form $f(z)$ of weight $k$ for the full modular group $SL(2,\ \mathbb{Z}).$ Here we call $f(z)$ a primitive cusp form if it is an eighenfunction of all Hecke operators simultaneously.
Let $\G\subset \mathrm{SL}_{2}(\R)$ be a cofinite Fuchsian subgroup, and let $i\infty$ be a cusp of $\G$. For $k\in\Z_{\geq 0}$, let $\Sk$ denote the complex vector space of cusp forms of weight-$k$, with respect to the Fuchsian subgroup $\G$. Let $f\in\Sk$ be a cusp form of weight-$k$, which is normalized, with respect to the Petersson inner-product on $\Sk$. For any $n\in\Z_{\geq 1}$, let $a_{n}$ denote the $n$-th Fourier coefficient of $f$ at $i\infty$. Then, for any $k\in\Z_{\geq 5}$, we show that \begin{align*} \big|a_{n}\big|=O_{f,\G}\big(n^{\frac{k-1}{2}+\frac{2}{k}}\big), \end{align*} …
In this article, we prove non-vanishing results for $L$-functions associated to holomorphic cusp forms of half-integral weight on average (over an orthogonal basis of Hecke eigenforms). This extends a result of W. Kohnen to forms of half-integral weight.
Let $S_k$ be the space of holomorphic cusp forms of weight $k$ with respect to $SL_2(\mathbb{Z})$. Let $f \in S_k$ be a normalized Hecke eigenform, $L_f(s)$ the $L$-function attached to the form $f$. In this paper we consider the distribution of zeros of $L_f(s)$ in the strip $σ\leq \Re s \leq 1$ for fixed $σ>1/2$ with respect to the imaginary part. We study estimates of \[ N_f(σ,T) = #\{ρ\in\mathbb{C} \mid L_f(ρ)=0, σ leq \Reρ\leq 1, 0 \leq \Imρ\leq T} \] for $1/2 \leq σ\leq1$ and large $T>0$. Using the methods of Karatsuba and Voronin we shall give another proof for Ivić's method.
We give a derivative version of the relative trace formula on PGL(2) studied in our previous work, and obtain a formula of an average of central values (derivatives) of automorphic $L$-functions for Hilbert cusp forms. As an application, we prove existence of Hilbert cusp forms with non-vanishing central values (derivatives) such that the absolute degrees of their Hecke fields are sufficiently large.
We estimate the sums \[ \sum_{c\leq x} \frac{S(m,n,c,χ)}{c}, \] where the $S(m,n,c,χ)$ are Kloosterman sums of half-integral weight on the modular group. Our estimates are uniform in $m$, $n$, and $x$ in analogy with Sarnak and Tsimerman's improvement of Kuznetsov's bound for the ordinary Kloosterman sums. Among other things this requires us to develop mean value estimates for coefficients of Maass cusp forms of weight $1/2$ and uniform estimates for $K$-Bessel integral transforms. As an application, we obtain an improved estimate for the classical problem of estimating the size of the error t…
Let $f$ be a Hecke-Maass cusp form for $\rm SL_2(\mathbb{Z})$ with Laplace eigenvalue $λ_f(Δ)=1/4+μ^2$ and let $λ_f(n)$ be its $n$-th normalized Fourier coefficient. It is proved that, uniformly in $α, β\in \mathbb{R}$, $$ \sum_{n \leq X}λ_f(n)e\left(αn^2+βn\right) \ll X^{7/8+\varepsilon}λ_f(Δ)^{1/2+\varepsilon}, $$ where the implied constant depends only on $\varepsilon$. We also consider the summation function of $λ_f(n)$ and under the Ramanujan conjecture we are able to prove $$ \sum_{n \leq X}λ_f(n)\ll X^{1/3+\varepsilon}λ_f(Δ)^{4/9+\varepsilon} $$ with the implied constant depending only …
The purpose of the article is to estimate the mean square of a squareroot length exponential sum of Fourier coefficients of a holomorphic cusp form.
In this article, for $n\geq 2$, we compute asymptotic, qualitative, and quantitative estimates of the Bergman kernel of Picard modular cusp forms associated to torsion-free, cocompact subgroups of $\mathrm{SU}\big((n,1),\mathbb{C}\big)$. The main result of the article is the following result. Let $Γ\subset \mathrm{SU}\big((2,1),\mathcal{O}_{K}\big)$ be a torsion-free subgroup of finite index, where $K$ is a totally imaginary field. Let $\mathcal{B}_Γ^{k}$ denote the Bergman kernel associated to the $\mathcal{S}_{k}(Γ)$, complex vector space of weight-$k$ cusp forms with respect to $Γ$. Let $\m…
The fourth moment of holomorphic Hecke cusp forms in shorter intervals
本文研究了较短区间内全纯 Hecke 尖形式四次矩的平均值，通过设定区间长度 $H=K^{3/4+c}$，改进了 Khan 的先前结果。
We prove the non-vanishing of special $L$-values of cuspidal automorphic forms on GL(2) twisted by Hecke characters of prime power orders and totally split prime power conductors. Main ingredients of the proof are estimating the Galois averages of fast convergent series expressions of special $L$-values and considering Shintani cone decomposition.
The summatory function of $t_j(n^2)$ is estimated, where $H_j(s) = \sum_{n=1}^\infty t_j(n)n^{-s}$ is the Hecke series of a non-holomorphic cusp form. The analogous problem of holomorphic cusp forms is also treated.
We derive a Motohashi-type formula for the cubic moment of central values of $L$-functions of level $q$ cusp forms twisted by quadratic characters of conductor $q$, previously studied by Conrey and Iwaniec and Young. Corollaries of this formula include Weyl-subconvex bounds for $L$-functions of weight two cusp forms twisted by quadratic characters, and estimates towards the Ramanujan-Petersson conjecture for Fourier coefficients of weight 3/2 cusp forms.
An additive problem in the Fourier coefficients of cusp forms
本文研究了任意级和特征的尖点形式傅里叶系数平移乘积之和的估计，并作为应用推广了前人工作，导出了其特征扭曲关联L函数在临界线上的次凸估计。
Metaplectic cusp forms and the large sieve
本文证明固定三次元辛尖点形式傅里叶系数在素数上求和的幂节省上界，通过新的双线性大筛估计和线性型估计克服分布水平2/3的瓶颈。
Let $Γ$ be a cocompact, discrete, and irreducible subgroup of $\mathrm{PSL}_{2}(\mathbb{R})^{n}$. Let $ν$ be a unitary character of $Γ$. For $k\in1\slash 2\,\mathbb{Z}$, let $\sknu$ denote the complex vector space of cusp forms of weight-$\tk=\k$ and nebentypus $ν^{2k}$ with respect to $Γ$. We assume that $ω_{X,ν}$, the line bundle of cusp forms of weight-$\tilde{1\slash 2}:=(1\slash 2,\ldots,1\slash2)$ with nebentypus $ν$ over $X$ exists. Let $\lbrace f_{1},\ldots,f_{j_{\tk}} \rbrace$ denote an orthonormal basis of $\sknu$. In this article, we show that as $k\rightarrow \infty$, the sum $\sum…
This note is concerned with the zeros of holomorphic Hecke cusp forms of large weight on the modular surface. The zeros of such forms are symmetric about three geodesic segments and we call those zeros that lie on these segments, real. Our main results give estimates for the number of real zeros as the weight goes to infinity.
We elaborate an explicit version of the relative trace formula on $\PGL(2)$ over a totally real number field for the toral periods of Hilbert cusp forms along the diagonal split torus. As an application, we prove (i) a spectral equidistribution result in the level aspect for Satake parameters of holomorphic Hilbert cusp forms weighted by central $L$-values, and (ii) a bound of quadratic base change $L$-functions for Hilbert cusp forms with a subconvex exponent in the weight aspect.
In this article, using methods from geometric analysis and theory of heat kernels, we derive qualitative estimates of automorphic cusp forms defined over quaternion algebras. Using which, we prove an average version of the holomorphic QUE conjecture. We then derive quantitative estimates of classical Hilbert modular cusp forms. This is a generalization of the results from [3] and [9] to higher dimensions.
Additive twists are important invariants associated to holomorphic cusp forms; they encode the Eichler--Shimura isomorphism and contain information about automorphic $L$-functions. In this paper we prove that central values of additive twists of the $L$-function associated to a holomorphic cusp form $f$ of even weight $k$ are asymptotically normally distributed. This generalizes (to $k\geq 4$) a recent breakthrough of Petridis and Risager concerning the arithmetic distribution of modular symbols. Furthermore we give as an application an asymptotic formula for the averages of certain 'wide' fam…
Applications of zero-free regions on averages and shifted convolution sums of Hecke eigenvalues
本文假设Vinogradov-Korobov型无零点区域与广义Ramanujan-Petersson猜想，建立了$SL(n,\mathbb{Z})$上Hecke-Maass尖形式傅里叶系数短和的非平凡上界，并应用于移位和均值及系数变号的条件性结果。
In this work we prove a prime number type theorem involving the normalised Fourier coefficients of holomorphic and Maass cusp forms, using the classical circle method. A key point is in a recent paper of Fouvry and Ganguly, based on Hoffstein-Ramakrishnan's result about the non-existence of the Siegel zeros for $GL(2)$ $L$-functions, which allows us to improve preceding estimates.
We consider large values of long linear exponential sums involving Fourier coefficients of holomorphic cusp forms. The sums we consider involve rational linear twists $e(nh/k)$ with sufficiently small denominators. We prove both pointwise upper bounds and bounds for the frequency of large values. In particular, the $k$-aspect is treated. As an application we obtain upper bounds for all the moments of the sums in question. We also give the asymptotics with the right main term for fourth moments. We also consider the mean square of very short sums, proving that on average short linear sums with …
An asymptotic formula of spectral average of central $L$-values on ${\bf GSp}(2)$ for square free levels
本文在 ${\f PGSp}_2$ 上发展新型相对迹公式，结合 Bessel 周期与 Rankin-Selberg 积分，证明了平方自由水平 Siegel 尖形式 Satake 参数的加权等分布定理。
Let $ψ$ be a function such that $ψ(x) \rightarrow \infty$ as $x \rightarrow \infty.$ Let $λ_{f}(n)$ be the $n$-th Hecke eigenvalue of a fixed holomorphic cusp form $f$ for $SL(2,\mathbb{Z}).$ We show that for any real valued function $h(x)$ such that $(\log X)^{2-2α} \ll h(X) =o(X),$ $$\sum_{n=x}^{x+h(X)} |λ_{f}(n)| \ll_{f} h(X)ψ(X)(\log X)^{α-1}$$ for all but $O_{f}( Xψ(X)^{-2})$ many integers $x\in [X,2X-h(X)],$ in which $α$ is the average value of $|λ_{f}(p)|$ over primes. We generalize this for $|λ_{f}(n)|^{2^{k}}$ for $k \in \mathbb{Z^{+}}.$
We prove equidistribution theorems for a family of holomorphic Siegel cusp forms of general degree in the level aspect. Our main contribution is to estimate unipotent contributions for general degree in the geometric side of Arthur's invariant trace formula in terms of Shintani zeta functions. Several applications including the vertical Sato-Tate theorem and low-lying zeros for standard $L$-functions of holomorphic Siegel cusp forms are discussed. We also show that the "non-genuine forms" which come from non-trivial endoscopic contributions by Langlands functoriality classified by Arthur are n…
On the Real Zeroes of Half-integral Weight Hecke Cusp Forms
研究半整数权Hecke自守形式在无穷尖点附近零点分布，证明零点数量随权增大而接近预期速率，并获得较弱的零点下界。
Let $k$ and $n$ be positive even integers. For a Hecke eigenform $h$ in the Kohnen plus subspace of weight $k-n/2+1/2$ for $\varGamma_0(4)$, let $I_n(h)$ be the Duke-Imamoglu-Ikeda lift of $h$ to the space of cusp forms of weight $k$ for $Sp_n(\mathbb Z)$. We then give an estimate of the Fourier coefficients of $I_n(h)$. It is better than the usual Hecke bound for the Fourier coefficients of a Siegel cusp form.
Approximations of $SL(3,\mathbb{Z})$ Hecke-Maass $L$-Functions by short Dirichlet polynomials
本文通过短Dirichlet多项式近似SL(3,Z) Hecke-Maass L函数，利用Kuznetsov迹公式方法，展示了L函数倒数的平均近似性。
Let $A_f(1,n)$ be the normalized Fourier coefficients of a $GL(3)$ Maass cusp form $f$ and let $a_g(n)$ be the normalized Fourier coefficients of a $GL(2)$ cusp form $g$. Let $λ(n)$ be either $A_f(1,n)$ or the triple divisor function $d_3(n)$. It is proved that for any $ε>0$, any integer $r\geq 1$ and $r^{5/2}X^{1/4+7δ/2}\leq H\leq X$ with $δ>0$, $$ \frac{1}{H}\sum_{h\geq 1}W\left(\frac{h}{H}\right) \sum_{n\geq 1}λ(n)a_g(rn+h)V\left(\frac{n}{X}\right)\ll X^{1-δ+ε}, $$ where $V$ and $W$ are smooth compactly supported functions, and the implied constants depend only on the associated forms and $…
On the density hypothesis for $L$-functions associated with holomorphic cusp forms
本文改进偶整权尖点 Hecke 特征形式对应 $L$-函数的零点密度估计，将有效范围推进至 $\sigma\geq 1407/1601$，核心在于结合 zeta 混合矩等工具改进 Dirichlet 多项式大值估计。
We give an exact formula of the average of adjoint $L$-functions of holomorphic Hilbert cusp forms with a fixed weight and a square-free level, which is a generalization of Zagier's formula known for the case of elliptic cusp forms on ${\rm SL}_2(\mathbb{Z})$. As an application, we prove that the Satake parameters of Hilbert cusp forms with a fixed weight and with growing square-free levels are equidistributed in an ensemble constructed by values of the adjoint $L$-functions.
We compute the quantum variance of holomorphic cusp forms on the vertical geodesic for smooth, compactly supported test functions. The variance is related to an averaged shifted-convolution problem that we evaluate asymptotically. We encounter an off-diagonal term that matches exactly with a certain diagonal term, a feature reminiscent of moments of $L$-functions.
Let $f$ be a primitive Maass cusp form for a congruence subgroup $Γ_0(D) \subset $ SL($2,\mathbb{Z}$) and $λ_f(n)$ its $n$-th Fourier coefficient. In this paper it is shown that with knowledge of only finitely many $λ_f(n)$ one can often solve for the level $D$, and in some cases, estimate the Laplace eigenvalue to arbitrarily high precision. This is done by analyzing the resonance and rapid decay of smoothly weighted sums of $λ_f(n)e(αn^β)$ for $X \leq n \leq 2X$ and any choice of $α\in \mathbb{R}$, and $β>0$. The methods include the Voronoi summation formula, asymptotic expansions of Bessel …
Uniform sup-norm bounds on average for Siegel cusp forms
本文利用热核方法，给出Siegel尖点形式的加权正交基平方和的上确界范数的平均上界，并证明该界在有限指数子群变化下具有一致性。
On the Rankin-Selberg problem in families
本文在全纯模形式和Hecke-Maass尖点形式族的短区间上研究Rankin-Selberg问题，对前者假设平均Lindelöf界，对后者不作假设。
Let F be an orthonormal basis of weight 2 cusp forms on Gamma_0(N). We show that various weighted averages of special values L(f \tensor chi, 1) over f in F are equal to 4 pi + O(N^{-1 + epsilon}). A previous result of Duke gives an error term of O(N^{-1/2} log N). The bound here is used in the author's paper "Galois representations attached to Q-curves and the generalized Fermat equation A^4 + B^2 = C^p," (to appear, Amer. J. Math.) to show that certain spaces of cuspforms arising there contain forms whose L-functions have nonvanishing special value. Version of May 2005: Nathan Ng found an er…
Quantum variance for cubic moment of Hecke--Maass cusp forms and Eisenstein series
本文分别给出了Hecke--Maass尖点形式和Eisenstein级数三次矩方差的上界，基于大筛不等式和Lindelöf均值型界，并建立了特殊点附近GL(2)相关L-函数四次矩和八次矩的精确上界。
Triple Correlation Sums of Coefficients of Cusp Forms
本文研究全纯尖点形式 Fourier 系数的三重平移关联和，给出无条件渐近估计及 Riemann 假设下的加强，并据此证明无穷多个三项等差数列上系数乘积非零。
A formula for the dimension of the space of cuspidal modular forms on $Γ_0(N)$ of weight $k$ ($k\ge2$ even) has been known for several decades. More recent but still well-known is the Atkin-Lehner decomposition of this space of cusp forms into subspaces corresponding to newforms on $Γ_0(d)$ of weight $k$, as $d$ runs over the divisors of $N$. A recursive algorithm for computing the dimensions of these spaces of newforms follows from the combination of these two results, but it would be desirable to have a formula in closed form for these dimensions. In this paper we establish such a closed-for…
We study low-lying zeros of $L$-functions attached to holomorphic cusp forms of level $1$ and large weight. In this family, the Katz--Sarnak heuristic with orthogonal symmetry type was established in the work of Iwaniec, Luo and Sarnak for test functions $ϕ$ satisfying the condition supp$(\widehat ϕ) \subset(-2,2)$. We refine their density result by uncovering lower-order terms that exhibit a sharp transition when the support of $\widehat ϕ$ reaches the point $1$. In particular the first of these terms involves the quantity $\widehat ϕ(1)$ which appeared in previous work of Fouvry--Iwaniec and…
Large deviations for values of $L$-functions attached to cusp forms in the level aspect
本文研究了素数级全纯尖点形式族中自守$L$-函数值的分布，证明了相关密度函数的渐近公式，并以此估计$L$-函数的大值。
In this paper, we give an optimal estimate of an average of Hurwitz class numbers. As an application, we give an equidistribution result of the family $\{\frac{t}{2q^{ν/2}} \ | \ ν\in \mathbb{N}, t \in \mathbb{Z}, |t|<2q^{ν/2}\}$ with $q$ prime, weighted by Hurwitz class numbers. This equidistribution produces many asymptotic relations among Hurwitz class numbers. Our proof relies on the resolvent trace formula of Hecke operators on elliptic cusp forms of weight $k\ge 2$.
We establish the Sato-Tate equidistribution of Hecke eigenvalues on average for families of Hecke--Maass cusp forms on SL(n,R)/SO(n). For each of the principal, symmetric square and exterior square L-functions we verify that the families are essentially cuspidal and deduce the level distribution with restricted support of the low-lying zeros. We also deduce average estimates toward Ramanujan.
Sums of Cusp Form Coefficients Along Quadratic Sequences
本文通过分析移位卷积和，改进了尖点形式系数沿二次序列求和的渐近误差界，并由附录给出更强的谱平均估计。
On the number of zeros of $L-$functions attached to cusp forms of half-integral weight
本文扩展了关于半整数权尖点形式 $L-$ 函数在临界线上零点数量的研究，通过发展 de la Valée Poussin 方法变体并利用 Lekkerkerker 变体改进了零点数量的下界估计。
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