Giacomo Cherubini, Alberto Perelli
We show that for a positive proportion of Laplace eigenvalues $λ_j$ the associated Hecke-Maass $L$-functions $L(s,u_j)$ approximate with arbitrary precision any target function $f(s)$ on a closed disc with center in $3/4$ and radius $r<1/4$. The main ingredients in the proof are the spectral large sieve of Deshouillers-Iwaniec and Sarnak's equidistribution theorem for Hecke eigenvalues.
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We construct a family of harmonic Maass forms of polynomial growth of any level corresponding to any cusp whose shadows are Eisenstein series of integral weight. We further consider Dirichlet series attached to a harmonic Maass form of polynomial growth, study its analytic properties, and prove an analogue of Weil's converse theorem.
We investigate the relations for $L$-functions satisfying certain functional equation, summationa formulas of Voronoi-Ferrar type and Maass forms of integral and half-integral weight. Summation formulas of Voronoi-Ferrar type can be viewed as an automorphic property of distribution vectors of non-unitary principal series representations of the double covering group of $SL(2)$. Our goal is converse theorems for automorphic distributions and Maass forms of level $N$ characterizing them by analytic properties of the associated $L$-functions. As an application of our converse theorems, we construc…
Weil's converse theorem for Maass forms and cancellation of zeros
本文证明了允许亚纯扭曲的 Maass 形式 Weil 型逆定理，并据此推出 Maass 新形式对称平方 L-函数与 Riemann zeta 函数之商有无穷多个极点。
We formulate a conjectured orthogonality relation between the Fourier coefficients of Maass forms on PGL(N) for N>=2. Based on the work of Goldfeld-Kontorovich and Blomer for N=3, and on our conjecture for N>=4, we prove a weighted vertical equidistribution theorem (with respect to the generalized Sato-Tate measure) for the Satake parameter of Maass forms at a finite prime. For N=3, the rate of convergence for the equidistribution theorem is obtained.
Using ideas of Ramakrishnan, we consider the icosahedral analogue of the theorems of Sarnak and Brumley on Hecke-Maass newforms with Fourier coefficients in a quadratic order. Although we are unable to conclude the existence of an associated Galois representation in this case, we show that one can deduce some implications of such an association, including weak automorphy of all symmetric powers and the value distribution of Fourier coefficients predicted by the Chebotarev density theorem.
L-series of harmonic Maass forms and a summation formula for harmonic lifts
本文引入了与调和Maass形式相关的L-级数并证明其函数方程，建立了该L-级数的逆定理，并作为应用证明了给定尖点形式的调和提升全纯部分的求和公式。
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.
Let $j(z)$ be the modular $j$-invariant function. Let $τ$ be an algebraic number in the complex upper half plane $\mathbb{H}$. It was proved by Schneider and Siegel that if $τ$ is not a CM point, i.e., $[\mathbb{Q}(τ):\mathbb{Q}]\neq2$, then $j(τ)$ is transcendental. Let $f$ be a harmonic weak Maass form of weight $0$ on $Γ_0(N)$. In this paper, we consider an extension of the results of Schneider and Siegel to a family of values of $f$ on Hecke orbits of $τ$. For a positive integer $m$, let $T_m$ denote the $m$-th Hecke operator. Suppose that the coefficients of the principal part of $f$ at t…
We prove a uniform estimate for sums of Hecke--Maass eigenvalues squared over primes in short intervals that can be regarded as an analogue of Hoheisel's classical prime number theorem for all real analytic cusp forms. Our argument is modelled after our treatment of Linnik's least prime number theorem for arithmetic progressions (Tata LN 72) and depends on recent works in the theory of automorphic representations. We stress that constants in the present work, including those implicit, are all universal and effectively computable, although we pay no particular attention to numerical precision. …
Low-lying zeros in families of Maass form L-functions: an extended density theorem
本文研究素数水平Maass形式L函数族低零点密度，将可容许支撑从(-3/2,3/2)扩展到(-15/8,15/8)，并假设大密度猜想可至(-2,2)，方法基于Kutznetsov迹公式、大筛法及四次矩界。
According to Waldspurger's theorem, the coefficients of half-integral weight eigenforms are given by central critical values of twisted Hecke L-functions, and therefore by periods. Here we prove that the coefficients of weight 1/2 harmonic Maass forms are determined by periods of algebraic differentials of the third kind on modular and elliptic curves.
We consider sign changes of Fourier coefficients of Hecke-Maass cusp forms for the group $\mathrm{SL}_3(\mathbb Z)$. When the underlying form is self-dual, we show that there are $\gg_\varepsilon X^{5/6-\varepsilon}$ sign changes among the coefficients $\{A(m,1)\}_{m\leq X}$ and that there is a positive proportion of sign changes for many self-dual forms. Similar result concerning the positive proportion of sign changes also hold for the real-valued coefficients $A(m,m)$ for generic $\mathrm{GL}_3$ cusp forms, a result which is based on a new effective Sato-Tate type theorem for a family of $\…
We investigate so-called "higher" Siegel theta lifts on Lorentzian lattices in the spirit of Bruinier-Ehlen-Yang and Bruinier-Schwagenscheidt. We give a series representation of the lift in terms of Gauss hypergeometric functions, and evaluate the lift as the constant term of a Fourier series involving the Rankin-Cohen bracket of harmonic Maass forms and theta functions. Using the higher Siegel lifts, we obtain a vector-valued analogue of Mertens' result stating that the Rankin--Cohen bracket of the holomorphic part of a harmonic Maass form of weight $\frac{3}{2}$ and a unary theta function, p…
Linnik problem for Maass--Hecke cuspforms and effective multiplicity one theorem
研究Maass-Hecke尖形式的Linnik问题与有效重数一性问题，通过新型对称平方谱大筛不等式等方法，改进了1991年Huntley的相关界，还讨论了向任意数域上的推广。
We explicitly construct cusp forms on the orthogonal group of signature $(1,8n+1)$ for an arbitrary natural number $n$ as liftings from Maass cusp forms of level one. In our previous works, the fundamental tool to show the automorphy of the lifting was the converse theorem by Maass. In this paper, we use the Fourier expansion of the theta lifts by Borcherds instead. We also study cuspidal representations generated by such cusp forms and show that they are irreducible and that all of their non-archimedean local components are non-tempered while the archimedean component is tempered, if the Maas…
L-Series for Vector-Valued Weakly Holomorphic Modular Forms and Converse Theorems
本文利用拉普拉斯变换引入弱全触模形式的L-级数并给出函数方程，进而确定了向量值调和弱Maass形式、Jacobi形式及半整数权椭圆模形式的逆定理。
Using the Kuznetsov formula, we prove several density theorems for exceptional Hecke and Laplacian eigenvalues of Maass cusp forms of weight 0 or 1 for the congruence subgroups $Γ_0(q)$, $Γ_1(q)$, and $Γ(q)$. These improve and extend upon results of Sarnak and Huxley, who prove similar but slightly weaker results via the Selberg trace formula.
We give a construction of a new $p$-adic Maass-Shimura operator defined on an affinoid subdomain of the preperfectoid $p$-adic universal cover $\mathcal{Y}$ of a modular curve $Y$. We define a new notion of $p$-adic modular forms as sections of a certain sheaf $\mathcal{O}_Δ$ of "nearly rigid functions" which transform under the action of subgroups of the Galois group $\mathrm{Gal}(\mathcal{Y}/Y)$ by $\mathcal{O}_Δ^{\times}$-valued weight characters. This extends Katz's notion of $p$-adic modular forms as functions on the Igusa tower $Y^{\mathrm{Ig}}$; indeed we may recover Katz's theory by re…
Limit theorem for the hybrid joint universality theorem on zeta and $L$-functions
本文针对 zeta 函数与 $L$-函数的混合联合普适性定理，克服 Bagchi 方法适配困难，首次给出其极限定理的概率证明。
Selberg's Central Limit Theorem for $L$-functions of level aspect, II
该研究在数论领域证明了与Hecke-Maass尖点形式相关的GL(3) L-函数的Selberg中心极限定理，同时证实了这类自守L-函数的独立性。
We prove joint universality theorems on the half plane of absolute convergence for general classes of Dirichlet series with an Euler-product, where in addition to vertical shifts we also allow scaling. This generalizes our recent joint universality results for Dirichlet $L$-functions. In contrast to classical universality, we do not need that the Dirichlet series in question have an analytic continuation beyond their region of absolute convergence. Also we may allow weaker orthogonality conditions for pairs of Dirichlet series than in the previous joint universality results of Lee-Nakamura-Pań…
In this paper, we prove the universality theorem for the iterated integrals of the logarithm of the Riemann zeta-function on some line parallel to the real axis.
Universality theorems of the Selberg zeta functions for arithmetic groups
本文将 Selberg zeta 函数的普适性定理从模群及其主同余子群进一步推广到模群的一般子群和由不定四元数代数导出的余紧算术群，并得到了比已有结果更宽的成立区域。
A GL(3) converse theorem via a "beyond endoscopy" approach
该研究通过受朗兰兹‘超越内窥镜法’启发的技术，给出GL(3)上Maass形式逆定理新证明，首次在高阶情形实现相关想法。
Arithmetic Springer theorem and $n$-universality under field extensions
基于BONGs理论证明一般二进局部域上二次型的整及相对整旋量范数的范数原理，建立不定二次型的算术版Springer定理，解决任意局部域上n-普遍性的提升问题。
Effective hybrid joint universality for Dirichlet $L$-functions and its application
本文研究了Dirichlet L-函数的混合联合普遍性定理，并应用于估计Hurwitz ζ-函数的普遍性定理的下密度下界。
Joint Universality for the Riemann zeta-function with general shifts
本文采用不同方法，将Riemann zeta函数一般位移的普遍性定理推广至联合普遍性情形，扩展了Andersson等人的研究结果。
The property of zeta-functions on mixed joint universality in the Voronin's sense states that any two holomorphic functions can be approximated simultaneously with accuracy $ε>0$ by suitable vertical shifts of the pair consisting from the Riemann zeta- and Hurwitz zeta-functions. In [1], it was shown rather general result, i.e., an approximating pair was composed of the Matsumoto zeta-functions' class and the periodic Hurwitz zeta-function. In this paper, we prove that this set of shifts has a strict positive density for all but at most countably many $ε>0$. Also, we give the concluding remark…
Let $0<r<1/4$, and $f$ be a non-vanishing continuous function in $|z|\leq r$, that is analytic in the interior. Voronin's universality theorem asserts that translates of the Riemann zeta function $ζ(3/4 + z + it)$ can approximate $f$ uniformly in $|z| < r$ to any given precision $\varepsilon$, and moreover that the set of such $t \in [0, T]$ has measure at least $c(\varepsilon) T$ for some $c(\varepsilon) > 0$, once $T$ is large enough. This was refined by Bagchi who showed that the measure of such $t \in [0,T]$ is $(c(\varepsilon) + o(1)) T$, for all but at most countably many $\varepsilon > …
The second moment of symmetric square L-functions over Gaussian integers
本文证明了高斯整数上Maass形式对称平方L函数二阶矩的新上界，并结合已有结果改进了Picard流形素数测地线定理的误差项。
Discrete universality, continuous universality and hybrid universality are equivalent
本研究针对Sourmelidis关于ζ函数离散与连续泛用性等价证明存在条件限制、单向推导的不足，无条件证明二者及混合泛用性的等价性，且证明不依赖Dirichlet级数表示，更具一般性。
The universality of symmetric power L-functions and their Rankin-Selberg L-functions
建立了自守形式的前四个对称幂L函数及其相应的Rankin-Selberg L函数的普适性定理。这推广了Laurincikas & Matsumoto和Matsumoto的一些结果.
Universality for the iterated integrals of logarithms of $L$-functions in the Selberg class
本文证明了在平行于实轴的某直线上，Selberg类中$L$-函数对数的迭代积分满足普适性定理。
Uniqueness of the 2-universality Criterion
Kim，Kim和Oh给出了正定整数矩阵二次型的2-普适性的一个最小判据。我们证明了这个2-普适性准则在Conway-Schneeberger 15定理的唯一性意义上是唯一的。
We use subsequence and moving average ergodic theorems applied to Boole's transformation and its variants and their invariant measures on the real line to give new characterisations of the Lindelh{ö}f Hypothesis and the Riemann hypothesis. These ideas are then used to study the value distribution of Dirichlet L series, and the zeta functions of Dedekind, Hurwitz and Riemann and their derivatives. This builds on earlier work of R. L. using Birkhoff's ergodic theorem and probability theory.
Continuous and discrete universality of zeta-functions: Two sides of the same coin?
本文借助线性动力学建立ζ函数连续与离散普适性之间的联系，并由连续情形直接推出文献中的多种离散普适性定理。
Discrete universality for Matsumoto zeta-functions and the nontrivial zeros of the Riemann zeta-function
本文将 Garunkštis 等人关于黎曼 zeta 函数经非平凡零点筛选的离散普适性定理推广到松本 zeta 函数。
In this paper, we study the set of positive integers that characterize the universality of $m$-gonal form.
On the value distribution of the Riemann zeta-function with general shifts
研究一般位移下黎曼ζ函数的值分布，通过建立偏差估计、证明普遍性定理及获得定量定理，在对数幂位移方面取得成果，此前一般位移普遍性结果未涵盖。
Notes on Universality in Short Intervals and Exponential Shifts
本文改进了Antanas Laurinčikas关于Riemann zeta-function在短区间内的普遍性定理，涉及区间长度，并证明平移可具有指数增长，该研究源于Laurinčikas提出的两个问题。
By elaborating a two-dimensional Selberg sieve with asymptotics and equidistributions of Kloosterman sums from $\ell$-adic cohomology, as well as a Bombieri--Vinogradov type mean value theorem for Kloosterman sums in arithmetic progressions, it is proved that for any given primitive Hecke--Maass cusp form of trivial nebentypus, the eigenvalue of the $n$-th Hecke operator does not coincide with the Kloosterman sum $\mathrm{Kl}(1,n)$ for infinitely many squarefree $n$ with at most $100$ prime factors. This provides a partial negative answer to a problem of Katz on modular structures of Kloosterm…
We prove a variant of the Lavrentiev's approximation theorem that allows us to approximate a continuous function on a compact set K in C without interior points and with connected complement, with polynomial functions that are nonvanishing on K. We use this result to obtain a version of the Voronin universality theorem for compact sets K, without interior points and with connected complement where it is sufficient that the function is continuous on K and the condition that it is nonvanishing can be removed. This implies a special case of a criterion of Bagchi, which in the general case has bee…
We study analytic properties of the pair consisting of a rather general form of zeta-function with an Euler product and a periodic Hurwitz zeta-function with a transcendental parameter. We first survey briefly previous results, and then investigate the mixed joint discrete value distribution and prove the mixed joint discrete universality theorem for these functions, in the case when common differences of relevant arithmetic progressions are not necessarily the same. Also some generalizations are given. For this purpose, certain arithmetic conditions on the common differences are necessary.
Universality of the zeta function in short intervals
该研究改进了黎曼ζ函数的短区间普适性定理，在黎曼假设下证明了显式参数B对应的短区间普适性，无条件下证明了相关实数集合具有正上密度。
We prove a version of Bagchi's Theorem and of Voronin's Universality Theorem for family of primitive cusp forms of weight $2$ and prime level, and discuss under which conditions the argument will apply to general reasonable family of automorphic $L$-functions.
We prove a variant of the Mergelyan approximation theorem that allows us to approximate functions that are analytic and nonvanishing in the interior of a compact set K with connected complement, and whose interior is a Jordan domain, with nonvanishing polynomials. This result was proved earlier by the author in the case of a compact set K without interior points, and independently by Gauthier for this case and the case of strictly starlike compact sets. We apply this result on the Voronin universality theorem for compact sets K of this type, where the usual condition that the function is nonva…
The Highly Oscillatory Behavior of Automorphic Distributions for SL(2)
SL（2）的自守分布作为模形式的边界值出现，并且以更微妙的方式从马斯形式出现。在权为1的模形式或马斯形式的情况下，自守分布具有连续的一阶反导数。我们回顾早先的结果，我们之一的持有人连续性的这些连续函数，并将它们与其他作者的结果，这涉及到一个推广的经典定理傅里叶级数的S。伯恩斯坦和哈迪-利特尔伍德然后，我们证明了反导数是不可微的所有无理点，以及所有，或在某些情况下，一些有理点。我们包括几个这些功能的图形，它清楚地显示了高度的振荡。我们的调查的动机部分由“黎曼不可微函数”，也被称为“维尔斯特拉斯函数”的性质。
Beyond the spherical sup-norm problem
本文针对极大紧子群非交换时的非球面 Maass 形式提出并求解了 SL_2(C) 算术商上的 sup 范数问题，达到亚 Wigner 指数，并发展了广义球面函数的解析理论与矩阵计数问题。
We prove a mixed joint discrete universality theorem for a Matsumoto zeta-function $φ(s)$ (belonging to the Steuding subclass) and a periodic Hurwitz zeta-function $ζ(s,α;{\mathfrak{B}})$. For this purpose, certain independence condition for the parameter $α$ and the minimal step of discrete shifts of these functions is assumed. This paper is a continuation of authors' works [12] and [arXiv:1601.03795].
Voronin Universality in several complex variables
本文证明了 $\mathbb{C}^n$ 中带有理或超越参数的多重 Hurwitz zeta 函数满足 Voronin 普适性定理，并由此给出首个多变量普适 Dirichlet 级数。
The goal of this paper is to explain certain experimentally observed properties of the (cuspidal) spectrum and its associated automorphic forms (Maass waveforms) on the congruence subgroup $Γ_{0}(9)$. The first property is that the spectrum possesses multiplicities in the so-called new part, where it was previously believed to be simple. The second property is that the spectrum does not contain any "genuinely new" eigenvalues, in the sense that all eigenvalues of $Γ_{0}(9)$ appear in the spectrum of some congruence subgroup of lower level. The main theorem in this paper gives a precise decompo…
We prove a new case of mixed discrete joint universality theorem on approximation of certain target couple of analytic functions by the shifts of a pair consisting of the function belonging to wide class of Matsumoto zeta-functions and the periodic Hurwitz zeta-function. We work under the condition that the common difference of arithmetical progression satisfies a certain rational condition and the parameter is a transcendental number. The essential difference from the result in our previous article is that here we do not study the class of partial zeta-functions, but work with the class of th…
We prove that the Voronin universality theorem for the Riemann zeta-function extends to the line Re(s)=1 if in addition to vertical shifts we also allow scaling and adding a sufficiently large constant.
Universality theorems for zeros of random real polynomials with fixed coefficients
该研究证明了含固定系数的随机首一实多项式恰有m个实零点的概率渐近公式，拓展了全随机系数情形的普遍性结果，并推导了n次代数整数实伽罗瓦共轭元计数的数论推论。
We survey some of the universality properties of the Riemann zeta function $ζ(s)$ and then explain how to obtain a natural quantization of Voronin's universality theorem (and of its various extensions). Our work builds on the theory of complex fractal dimensions for fractal strings developed by the second author and M. van Frankenhuijsen in \cite{La-vF4}. It also makes an essential use of the functional analytic framework developed by the authors in \cite{HerLa1} for rigorously studying the spectral operator $\mathfrak{a}$ (mapping the geometry onto the spectrum of generalized fractal strings)…
Gaussian Universality of Products Over Split Reductive Groups and the Satake Isomorphism
该研究利用Satake同构与Hall-Littlewood多项式，刻画非阿基米德域上分裂约化群乘积的奇异数与角元，推广已有结果并证明其最小分离性，推导相关概率极限定理，建立代数分解与概率极限的统一框架。
Effective Weak Universality in Short Intervals
本文证明了Riemann zeta函数在短区间中的有效普适性定理，并假设Riemann假设下扩展了区间范围。
Quantative universality for cokernels of matrices with symmetries
本文通过量化方法证明了具有对称性的随机整数矩阵余核的 universality，解决了伍德关于有效收敛速率的问题。
Entropy of affine permutations and universality of affine atomic lengths
本文研究了有理解排列的熵，并证明其与基本权重和的原子长度一致。通过Hall定理证明了非负整数可作为熵或核心多重划分的大小。进一步给出任意主导权重的原子长度多项式表达，并探讨其普遍性。
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