We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the third degree, and therefore the implications for the distribution of primes in short intervals, are the same as those we obtained (in the first paper with this title) by using the simpler approximation ΛR(n). However, when λR(n) is used, the error in the singular series approximation is often much smaller than what ΛR(n) allows. Assuming the Generalized Riemann Hypothesis (GRH) for Dirichlet L-functions, we obtain an Ω±-result for the variation of the error term in the prime number theorem. Formerly, our knowledge under GRH was restricted to Ω-results for the absolute value of this variation. An important ingredient in the last part of this work is a recent result due to Montgomery and Soundararajan which makes it possible for us to dispense with a large error term in the evaluation of a certain singular series average. We believe that our results on the sums λR(n) and ΛR(n) can be employed in diverse problems concerning primes. 1.

This paper develops an analytic theory of Dirichlet series in several complex variables which possess sufficiently many functional equations. In the first two sections it is shown how straightforward conjectures about the meromorphic continuation and polar divisors of certain such series imply, as a consequence, precise asymptotics (previously conjectured via random matrix theory) for moments of zeta functions and quadratic L-series. As an application of the theory, in a third section, we obtain the current best known error term for mean values of cubes of central values of Dirichlet L-series. The methods utilized to derive this result are the convexity principle for functions of several complex variables combined with a knowledge of groups of functional equations for certain multiple Dirichlet series.

. For scattering on the modular surface and on the hyperbolic cylinder, we show that the solutions of the wave equations can be expanded in terms of resonances, despite the presence of trapping. Expansions of this type are expected to hold in greater generality but have been understood only in non-trapping situations. 1. Introduction In this note we give two examples for which we can obtain, on compact sets, an asymptotic expansion of solutions to the wave equation with smooth, compactly supported initial data, although there is trapping. The expansions are given in terms of resonances and they generalize the standard "separation of variables" expansions in terms of eigenvalues. The examples are the modular surface where we can use detailed information about the zeta function (see Fig.1 and Theorem 1) and the hyperbolic cylinder where the resonances are particularly simple (see Fig.2(b) and Theorem 2). Resonances or scattering poles are defined as poles of the meromorphic continuat...

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 well as non-self-dual ones. A self-contained description of our proof for SL2(Z)\H is included to convey the main new ideas. Heavy use is made of truncation and the Maass-Selberg relations. 1

this paper we will not need such a restriction. (5) The condition ` ! 1=2 turns out to be surprisingly important. Note that

Abstract. The convergence theory for the set of simultaneously ψ-approximable points lying on a planar curve is established. Our results complement the divergence theory developed in [1] and thereby completes the general metric theory for planar curves.

ABSTRACT. We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that, for any η> 0, a positive proportion of consecutive primes are within 1 + η times the average spacing between primes. 4 1.

We give a newheuristic for all of the main terms in the integral moments of various families of primitive L-functions. The results agree with previous conjectures for the leading order terms. Our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary, orthogonal, or symplectic matrices, where the moments are de ned by the appropriate group averages. This lends support to the idea that arithmetical L-functions have a spectral interpretation, and that their value distributions can be modeled using Random Matrix Theory. Numerical examples show good agreement with our conjectures.

A detailed discussion of semiclassical trace formulae is presented and it is demonstrated how a regularized trace formula can be derived while dealing only with finite and convergent expressions. Furthermore, several applications of trace formula techniques to quantum chaos are reviewed. Then local spectral statistics, measuring correlations among finitely many eigenvalues, are reviewed and a detailed semiclassical analysis of the number variance is given. Thereafter the transition to global spectral statistics, taking correlations among infinitely many quantum energies into account, is discussed. It is emphasized that the resulting limit distributions depend on the way one passes to the global scale. A conjecture on the distribution of the fluctuations of the spectral staircase is explained in this general context and evidence supporting the conjecture is discussed. 1 Lectures held at the 3rd International Summer School/Conference Let's face chaos through nonlinear dynamics at the Un...

Abstract. This paper studies the non-holomorphic Eisenstein series E(z, s) for the modular surface PSL(2, Z)\H, and shows that integration with respect to certain non-negative measures µ(z) gives meromorphic functions Fµ(s) that have all their zeros on the line ℜ(s) = 1 2. For the constant term a0(y, s) of the Eisenstein series the Riemann hypothesis holds for all values y ≥ 1, with at most two exceptional real zeros, which occur exactly for those y> 4πe −γ = 7.0555+. The Riemann hypothesis holds for all truncation integrals with truncation parameter T ≥ 1. At the value T = 1 this proves the Riemann hypothesis for a zeta function Z2,Q(s) recently introduced by Lin Weng, associated to rank 2 semistable lattices over Q. 1.