The Beurling-Nyman criterion for the Riemann hypothesis: numerical aspects. (Le critère de Beurling et Nyman pour l'hypothèse de Riemann: aspects numérique.)
The Haros-Farey sequence at two hundred years. A survey.
The joint universality and the generalized strong recurrence for Dirichlet L-functions
The k-functions in multiplicative number theory. I. On complex explicit formulae
The k-functions in multiplicative number theory. II. Uniform distribution of zeta zeros
The Lax-Phillips infinitesimal generator and the scattering matrix for automorphic functions
We study the infinitesimal generator of the Lax-Phillips semigroup of the automorphic scattering system defined on the Poincaré upper half-plane for SL₂(ℤ). We show that its spectrum consists only of the poles of the resolvent of the generator, and coincides with the poles of the scattering matrix, counted with multiplicities. Using this we construct an operator whose eigenvalues, counted with algebraic multiplicities (i.e. dimensions of generalized eigenspaces), are precisely the non-trivial zeros...
The n-level densities of low-lying zeros of quadratic Dirichlet L-functions
Previous work by Rubinstein and Gao computed the n-level densities for families of quadratic Dirichlet L-functions for test functions f̂₁, ..., f̂ₙ supported in , and showed agreement with random matrix theory predictions in this range for n ≤ 3 but only in a restricted range for larger n. We extend these results and show agreement for n ≤ 7, and reduce higher n to a Fourier transform identity. The proof involves adopting a new combinatorial perspective to convert all terms to a canonical form,...
The pair correlation of zeros of Dirichlet L-functions and primes in arithmetic progressions.
The prime number theorem in short intervals for automorphic L-functions
The product of linear forms in a field of series and the Riemann hypothesis for curves
Trace formula in noncommutative Geometry and the zeros of the Riemann zeta function
Triple correlation of the Riemann zeros
We use the conjecture of Conrey, Farmer and Zirnbauer for averages of ratios of the Riemann zeta function [11] to calculate all the lower order terms of the triple correlation function of the Riemann zeros. A previous approach was suggested by Bogomolny and Keating [6] taking inspiration from semi-classical methods. At that point they did not write out the answer explicitly, so we do that here, illustrating that by our method all the lower order terms down to the constant can be calculated rigourously...