Degree 1 elements of the Selberg class

Strong multiplicity one for the Selberg class

We study the problem of determining elements of the Selberg class by information on the coefficents of the Dirichlet series at the squares of primes, or information about the zeroes of the functions.

On the zeros of functions in the Selberg class

It is proved that under some suitable conditions, the degree two functions in the Selberg class have infinitely many zeros on the critical line.

An extension of Venkatesh's converse theorem to the Selberg class

We extend Venkatesh's proof of the converse theorem for classical holomorphic modular forms to arbitrary level and character. The method of proof, via the Petersson trace formula, allows us to treat arbitrary degree 2 gamma factors of Selberg class type.

Strong multiplicity one for the Selberg class

In this note we investigate the problem of determining elements of the Selberg class from their Dirichlet series coefficients at the primes. We show that this is possible when the degree is one, but in general need an additional weak hypothesis on the Dirichlet series coefficients at squares of primes.

Effective uniform approximation by $L$-functions in the Selberg class

Recently, Garunkštis, Laurinčikas, Matsumoto, J. & R. Steuding showed an effective universality-type theorem for the Riemann zeta-function by using an effective multi-dimensional denseness result of Voronin. We will generalize Voronin's effective result and their theorem to the elements of the Selberg class satisfying some conditions.

On the variance of sums of arithmetic functions over primes in short intervals and pair correlation for L-functions in the Selberg class

We establish the equivalence of conjectures concerning the pair correlation of zeros of $L$-functions in the Selberg class and the variances of sums of a related class of arithmetic functions over primes in short intervals. This extends the results of Goldston & Montgomery [7] and Montgomery & Soundararajan [11] for the Riemann zeta-function to other $L$-functions in the Selberg class. Our approach is based on the statistics of the zeros because the analogue of the Hardy-Littlewood conjecture for the auto-correlation of the arithmetic functions we consider is not available in general. One of o…

关于塞尔伯格（Selberg）类狄利克雷级数：小次数情形

On the Selberg class of Dirichlet series: small degrees

本文梳理了具有算术意义的狄利克雷级数研究中，学界围绕函数方程、欧拉乘积与黎曼假设的关联，s=1单极点函数的结构，以及自守形式梅林变换的逆定理这三方面形成的相关猜想。

On analogues of Mazur-Tate type conjectures in the Rankin-Selberg setting

We study the Fitting ideals over the finite layers of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ of Selmer groups attached to the Rankin--Selberg convolution of two modular forms $f$ and $g$. Inspired by the Theta elements for modular forms defined by Mazur and Tate in ``Refined conjectures of the Birch and Swinnerton-Dyer type'', we define new Theta elements for Rankin--Selberg convolutions of $f$ and $g$ using Loeffler--Zerbes' geometric $p$-adic $L$-functions attached to $f$ and $g$. Under certain technical hypotheses, we generalize a recent work of Kim--Kurihara on elliptic cu…

关于 Selberg 类的密度假设

On the Density Hypothesis for the Selberg class

本文针对 Selberg 类证明了一般零点密度定理，在次数不超过 2 且实部至少为 0.9 时验证了密度假设，其他情形则得到较弱结果。

Beyond the extended Selberg class: $1<d_F< 2$

We show that a class of Dirichlet series ${\mathfrak{A}}^{\#}$ that is much larger than the extended Selberg class ${\mathscr{S}}^{\#}$, and also contains the standard as well as the tensor product, exterior square and symmetric square $L$-functions of automorphic $L$-functions of $GL_n$ over number fields, does not have any elements of degrees between $1$ and $2$. The proof of our more general theorem is very different from the proof of Kaczorowski and Perelli for the class ${\mathscr{S}}^{\#}$, and is much shorter and simpler even in that case.

t 方面下自守 $L$-函数的 Selberg 中心极限定理

Selberg's Central Limit Theorem for automorphic $L$-functions in the $t$-aspect

该研究采用 Radziwiłł 和 Soundararajan 的方法证明了 2 次自守 $L$-函数的 Selberg 中心极限定理，还证明了本原全纯尖点形式序列对应的自守 $L$-函数具有独立性。

On the Correlations, Selberg Integral and Symmetry of Sieve Functions in Short Intervals

We study the arithmetic (real) function f=g*1, with g "essentially bounded" and supported over the integers of [1,Q]. In particular, we obtain non-trivial bounds, through f "correlations", for the "Selberg integral" and the "symmetry integral" of f in almost all short intervals [x-h,x+h], N<x<2N, beyond the "classical" level, up to level of distribution, say, lambda=log Q/log N < 2/3 (for enough large h). This time we don't apply Large Sieve inequality, as in our paper [C-S]. Precisely, our method is completely elementary.

The $\mathbb F_p$-Selberg Integral

We prove an $\mathbb F_p$-Selberg integral formula, in which the $\mathbb F_p$-Selberg integral is an element of the finite field $\mathbb F_p$ with odd prime number $p$ of elements. The formula is motivated by analogy between multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo $p$.

The function field Sathé-Selberg formula in arithmetic progressions and `short intervals'

We use a function field analogue of a method of Selberg to derive an asymptotic formula for the number of (square-free) monic polynomials in $\mathbb{F}_q[X]$ of degree $n$ with precisely $k$ irreducible factors, in the limit as $n$ tends to infinity. We then adapt this method to count such polynomials in arithmetic progressions and short intervals, and by making use of Weil's `Riemann hypothesis' for curves over $\mathbb{F}_q$, obtain better ranges for these formulae than are currently known for their analogues in the number field setting. Finally, we briefly discuss the regime in which $q$ t…

Classification of L-functions of degree 2 and conductor 1

We give a full description of the functions $F$ of degree 2 and conductor 1 in the general framework of the extended Selberg class. This is performed by means of a new numerical invariant $χ_F$, which is easily computed from the data of the functional equation. We show that the value of $χ_F$ gives a precise description of the nature of $F$, thus providing a sharp form of the classical converse theorems of Hecke and Maass. In particular, our result confirms, in the special case under consideration, the conjecture that the functions in the Selberg class are automorphic $L$-functions.

The $\mathbb F_p$-Selberg integral of type $A_n$

We prove an $\mathbb F_p$-Selberg integral formula of type $A_n$, in which the $\mathbb F_p$-Selberg integral is an element of the finite field $\mathbb F_p$ with odd prime number $p$ of elements. The formula is motivated by analogy between multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo $p$. For the type $A_1$ the formula was proved in a previous paper by the authors.

$\mathrm{GU}(2,2)$ 的双变量 Rankin-Selberg 积分与 $\mathrm{GSp}_4$ 的 5 次 $L$-函数

A two variable Rankin-Selberg integral for $\mathrm{GU}(2,2)$ and the degree 5 $L$-function of $\mathrm{GSp}_4$

本文提出了一种双变量 Rankin-Selberg 积分，用于表示 exterior square $L$-functions 的乘积，并作为 residue 得到了被二次特征扭曲的 $\mathrm{GSp}_4$ 5 次 $L$-函数的积分表示。

Value distribution for the derivatives of the logarithm of $L$-functions from the Selberg class in the half-plane of absolute convergence

In the present paper, we show that, for every $δ>0$, the function $(\log {\mathcal{L}}(s))^{(m)}$, where $m\in {\mathbb{N}} \cup \{ 0\}$ and ${\mathcal{L}} (s) := \sum_{n=1}^\infty a(n) n^{-s}$ is an element of the Selberg class ${\mathcal{S}}$, takes any value infinitely often in any strip $1<\Re(s) <1+δ$, provided $\sum_{p\leq x} |a (p)|^2 \sim κπ(x)$ for some $κ>0$. In particular, ${\mathcal{L}} (s)$ takes any non-zero value infinitely often in the strip $1<\Re(s)<1+δ$, and the first derivative of ${\mathcal{L}} (s)$ vanishes infinitely often.

Generations of correlation averages

The present paper is a dissertation on the possible consequences of a conjectural bound for the so-called \thinspace modified Selberg integral of the divisor function $d_3$, i.e. a discrete version of the classical Selberg integral, where $d_3(n)=\sum_{abc=n}1$ is attached to the Cesaro weight $1-|n-x|/H$ in the short interval $|n-x|\le H$. Mainly, an immediate consequence is a non-trivial bound for the Selberg integral of $d_3$, improving recent results of Ivić based on the standard approach through the moments of the Riemann zeta function on the critical line. We proceed instead with element…

关于三个Hermitian自守形式的Rankin-Selberg积分

On a Rankin-Selberg integral of three Hermitian cusp forms

本文研究了三个Hermitian自守形式的Rankin-Selberg积分，通过分析狄利克雷级数的欧拉乘积性质，探讨了其与自守形式之间的关系。

An alternative proof of the Dirichlet prime number theorem

Dirichlet's theorem on arithmetic progressions called as Dirichlet prime number theorem is a classical result in number theory. Atle Selberg\cite{Selberg} gave an elementary proof of this theorem. In this article we give an alternative proof of it based on a previous result of us. Also we get an estimation of the prime counting function in the special cases.

关于二维 $\mathbb F_p$-Selberg 积分的注记

Notes on $2D$ $\mathbb F_p$-Selberg integrals

本研究证明了依赖于多组正整数参数、取值于奇素数阶有限域的二维 $\mathbb F_p$-Selberg 积分公式，其灵感源于KZ方程的多维超几何解与模p约化后多项式解的相似性，丰富了有限域上积分公式的研究成果。

Beyond the extended Selberg class: $d_F\le 1$

We will introduce two new classes of Dirichlet series which are monoids under multiplication. The first class $\mathfrak{A}^{\#}$ contains both the extended Selberg class $\mathscr{S}^{\#}$ of Kaczorowski and Perelli as well as many $L$-functions attached to automorphic representations of ${\rm GL}_n({\mathbb A}_K)$, where ${\mathbb A}_K$ denotes the adèles over the number field $K$ (these representations need not be unitary or generic). This is in contrast to the class $\mathscr{S}^{\#}$ which is smaller and is known to contain, very few of these $L$-functions. The larger class is obtained by…

On the standard twist of the L-functions of half-integral weight cusp forms

The standard twist $F(s,α)$ of $L$-functions $F(s)$ in the Selberg class has several interesting properties and plays a central role in the Selberg class theory. It is therefore natural to study its finer analytic properties, for example the functional equation. Here we deal with a special case, where $F(s)$ satisfies a functional equation with the same $Γ$-factor of the $L$-functions associated with the cusp forms of half-integral weight; for simplicity we present our results directly for such $L$-functions. We show that the standard twist $F(s,α)$ satisfies a functional equation reflecting $…

$s=1$附近零点与$L$-函数在Selberg类中的常数项

Zeros near $s=1$ and the constant term of $L'/L$ for $L$-functions in the Selberg class

研究$L$-函数在$s=1$附近零点分布与常数项的关系，证明零点分布与常数项的约束条件相互影响，推广了数域狄利克雷ζ函数的研究。

Forbidden conductors of L-functions and continued fractions of particular form

In this paper we study the forbidden values of the conductor $q$ of the $L$-functions of degree 2 in the extended Selberg class by a novel technique, linking the problem to certain continued fractions and to their weight $w_q$. Our basic result states that if an $L$ function with conductor $q$ exists, then the weight $w_q$ is unique in a suitable sense. From this we deduce several results, both of theoretical and computational nature.

Minimal Degrees of Algebraic Numbers with respect to Primitive Elements

Given a number field $L$, we define the degree of an algebraic number $v \in L$ with respect to a choice of a primitive element of $L$. We propose the question of computing the minimal degrees of algebraic numbers in $L$, and examine these values in degree $4$ Galois extensions over $\mathbb{Q}$ and triquadratic number fields. We show that computing minimal degrees of non-rational elements in triquadratic number fields is closely related to solving classical Diophantine problems such as congruent number problem as well as understanding various arithmetic properties of elliptic curves.

A note on hyperquadratic elements of low algebraic degree

In different areas of discrete mathematics, a certain type of polynomials, having coefficients in a field K of finite characteristic, has been considered. The form and the degree of these polynomials, here called projective, are simply linked to the characteristic p of K. Roots of these projective polynomials are particular algebraic elements over K, called hyperquadratic. For a general algebraic element of degree d over K, we discuss the possibility of being hyperquadratic. Using a method of differential algebra, we obtain, for particular fields K = Fp, projective polynomials only having poly…

On unitary groups associated to division algebras of degree three

We show that the special unitary group associated to an involution of the second kind on a central division algebra of degree three does not contain hermitian or skew-hermitian elements. Especially, there are no reflections. For Albert's special cyclic presentation we show that the intersections of reasonable $S$-arithmetic subgroups with the two obvious maximal subfields consist of the trivial central elements.

一次曲线纤维与四度德尔皮佐表面上的垂直布雷尔元素

Genus one fibrations and vertical Brauer elements on del Pezzo surfaces of degree 4

本文研究四度德尔皮佐表面中一次曲线纤维的几何与算术性质，探讨垂直布雷尔元素的存在与影响。