Rational Points on the Sphere

Using only basic tools from the theory of modular forms, the rational points of bounded height on the sphere are counted and shown to be uniformly distributed. The more difficult case of points with a given height is also treated.     

The Generalized Prime Number Theorem for Automorphic L-Functions

Let π and π' be automorphic irreducible cuspidal representations of GLm (QA) and GLm', (QA), respectively, and L(s,π×(～π)') be the Rankin-Selberg L-function attached to π and π'. Without assuming the Generalized Ramanujan Conjecture (GRC), the author gives the generalized prime number theorem for L(s, π×(～π)') when π(=)π'. The result generalizes the corresponding result of Liu and Ye in 2007.     

带有尖形式Fourier系数的指数和估计

设λ_f/(n)是全模群Γ上权为k的全纯Hecke特征形f的第n个Fourier系数,Λ(n)是Mangoldt函数.本文得到了如下估计∑_(Xn≤2X)Λ(n)λ_f(n)e(n~(1/2)α)■f,αX~(5/6)(logX)~(13/2),(α0),改进了Zhao的结果。     

Real zeros of Eisenstein series and Rankin-Selberg L-functions

We prove that the Eisenstein series E(z, s) have no real zeroes for s ∈ (0, 1) when the value of the imaginary part of z is in the range $\tfrac{1}{5}$ < Im z < 4.94. For very large and very small values of the imaginary part of z, E(z, s) have real zeros in (½, 1), i.e. GRH does not hold for the Eisenstein series. Using these properties, we prove that the Rankin-Selberg L-function attached with the Ramanujan τ-function has no real zeros in the critical strip, except at the central point s = ½.     

Representations of integers by certain positive definite binary quadratic forms

We prove part of a conjecture of Borwein and Choi concerning an estimate on the square of the number of solutions to n=x ²+Ny ² for a squarefree integer N.      

Prime number theorems for Rankin-Selberg L-functions over number fields

In this paper we define a Rankin-Selberg L-function attached to automorphic cuspidal represen-tations of GLm(AE) × GLm (AF ) over cyclic algebraic number fields E and F which are invariant under the Galois action,by exploiting a result proved by Arthur and Clozel,and prove a prime number theorem for this L-function.     

The Generalized Prime Number Theorem for Automorphic L-Functions

Let π and π＇ be automorphic irreducible cuspidal representations of GLm（QA） and GLm′ （QA）, respectively, and L（s, π×π′） be the Rankin-Selberg L-function attached to π and π＇. Without assuming the Generalized Ramanujan Conjecture （GRC）, the author gives the generalized prime number theorem for L（s, π × π′） when π =π＇. The result generalizes the corresponding result of Liu and Ye in 2007.     

Certain series attached to an even number of elliptic modular forms

For j=1,…,n let f_j(z) and g_j(z) be holomorphic modular forms for such that f_j(z)g_j(z) is a cusp form. We define a series

     

The Life and Work of R.A. Rankin (1915–2001)

Robert Alexander Rankin, an eminent Scottish number theorist and, for several decades, one of the world's foremost experts in modular forms, died on January 27, 2001 in Glasgow at the age of 85. He was one of the founding editors of The Ramanujan Journal. For this and the next two issues of the The Ramanujan Journal, many well-known mathematicians have prepared articles in Rankin's memory. In this opening paper, we provide a short biography of Rankin and discuss some of his major contributions to mathematics. At the conclusion of this article, we provide a complete list of Rankin's doctoral students and a complete bibliography of all of Rankin's writings divided into five categories: (1) Research and Expository Papers; (2) Books; (3) Books Edited; (4) Obituaries; (5) Other Writings.     

An L-Function of Degree 27 for Spin9

This paper studies a Rankin-Selberg integral for a degree 27 L-function on Spin(9). It makes use of an Eisenstein series on the exceptional group F ₄.