We develop a partial trace formula which circumvents some technical difficulties in computing the Selberg trace formula for the quotient SL3(Z)\SL3(R)/SO3(R). As applications, we establish the Weyl asymptotic law for the discrete Laplace spectrum and prove that almost all of its cusp forms are tempered at infinity. The technique shows there are non-lifted cusp forms on SL3(Z)\SL3(R)/SO3(R) as w...

(Under the Riemann Hypothesis the stronger bound |ζ(1 + it)| ≥ c log log t holds.) From a modern point of view, the method of de la Vallée Poussin is based on Rankin-Selberg convolutions and a positivity argument (an effective version of Landau’s Lemma – see [HL94, Appendix]). It can be applied to any Rankin-Selberg L-function L(s, π1 ⊗ π2), provided that one of the πi’s is self-dual (cf. [Sar0...

We present the simplest possible example of the Rankin-Selberg method, namely for a pair of holomorphic modular forms for SL(2,Z), treated independently in 1939 by Rankin and 1940 by Selberg. (Rankin has remarked that the general idea came from his advisor and mentor, Ingham.) We also recall a proof of the analytic continuation of the relevant Eisenstein series. That is, we consider the simples...

Abstract We give a relation between the orders of potential non-trivial zeros Dirichlet L -functions and analytic properties certain series first introduced by Selberg.

We prove some value-distribution results for a class of L-functions with rational moving targets. The class contains Selberg class, as well as the Riemann-zeta function.

We derive bounds on the extremal singular values and the condition number of N × K, with N > K, Vandermonde matrices with nodes in the unit disk. Such matrices arise in many fields of applied mathematics and engineering, e.g., in interpolation and approximation theory, sampling theory, compressed sensing, differential equations, control theory, and line spectral estimation. The mathematical tec...

INTRODUCTION 4 Introduction Zeta functions encoding geometric information such as zeta functions of algebraic varieties over finite fields or zeta functions of finite graphs will loosely be called geometric zeta functions in the sequel. Sometimes the geometric situation gives one tools at hand to prove analytical continuation, functional equation and an adapted form of the Riemann hypothesis. T...

P rime numbers are the atoms of our mathematical universe. Euclid showed that there are infinitely many primes, but the subtleties of their distribution continue to fascinate mathematicians. Letting p(n) denote the number of primes p B n, Gauss conjectured in the early nineteenth century that pðnÞ#n=lnðnÞ. In 1896, this conjecture was proven independently by Jacques Hadamard and Charles de la V...

The notion of a tower of Rankin-Selberg integrals was introduced in [G-R]. To recall this notion, let G be a reductive group defined over a global field F . Let G denote the L group of G. Let ρ denote a finite dimensional irreducible representation of G. Given an irreducible generic cuspidal representation of G(A), we let L(π, ρ, s) denote the partial L function associated with π and ρ. Here s ...