Universality theorems of the Selberg zeta functions for arithmetic groups

After Voronin proved the universality theorem of the Riemann zeta function in the 1970s, universality theorems have been proposed for various zeta and L-functions. Drungilas-Garunkstis-Kacenas' work at 2013 on the universality theorem of the Selberg zeta function for the modular group is one of them and is probably the first universality theorem of the zeta function of order greater than one. Recently, Mishou (2021) extended it by proving the joint universality theorem for the principal congruence subgroups. In the present paper, we further extend these works by proving the (joint) universality theorem for subgroups of the modular group and co-compact arithmetic groups derived from indefinite quaternion algebras, which is available in the region wider than the regions in the previous two works.Comment: 20 page

A method for calculating spectral statistics based on random-matrix universality with an application to the three-point correlations of the Riemann zeros

We illustrate a general method for calculating spectral statistics that combines the universal (Random Matrix Theory limit) and the non-universal (trace-formula-related) contributions by giving a heuristic derivation of the three-point correlation function for the zeros of the Riemann zeta function. The main idea is to construct a generalized Hermitian random matrix ensemble whose mean eigenvalue density coincides with a large but finite portion of the actual density of the spectrum or the Riemann zeros. Averaging the random matrix result over remaining oscillatory terms related, in the case of the zeta function, to small primes leads to a formula for the three-point correlation function that is in agreement with results from other heuristic methods. This provides support for these different methods. The advantage of the approach we set out here is that it incorporates the determinental structure of the Random Matrix limit.Comment: 22 page

Riemann zeta function and quantum chaos

A brief review of recent developments in the theory of the Riemann zeta function inspired by ideas and methods of quantum chaos is given.Comment: Lecture given at International Conference on Quantum Mechanics and Chaos, Osaka, September 200

Developments in Random Matrix Theory

In this preface to the Journal of Physics A, Special Edition on Random Matrix Theory, we give a review of the main historical developments of random matrix theory. A short summary of the papers that appear in this special edition is also given.Comment: 22 pages, Late

Group entropies, correlation laws and zeta functions

The notion of group entropy is proposed. It enables to unify and generalize many different definitions of entropy known in the literature, as those of Boltzmann-Gibbs, Tsallis, Abe and Kaniadakis. Other new entropic functionals are presented, related to nontrivial correlation laws characterizing universality classes of systems out of equilibrium, when the dynamics is weakly chaotic. The associated thermostatistics are discussed. The mathematical structure underlying our construction is that of formal group theory, which provides the general structure of the correlations among particles and dictates the associated entropic functionals. As an example of application, the role of group entropies in information theory is illustrated and generalizations of the Kullback-Leibler divergence are proposed. A new connection between statistical mechanics and zeta functions is established. In particular, Tsallis entropy is related to the classical Riemann zeta function.Comment: to appear in Physical Review

Bagchi's Theorem for families of automorphic forms

We prove a version of Bagchi's Theorem and of Voronin's Universality Theorem for family of primitive cusp forms of weight $2$ and prime level, and discuss under which conditions the argument will apply to general reasonable family of automorphic $L$-functions.Comment: 15 page

Fractal Complex Dimensions, Riemann Hypothesis and Invertibility of the Spectral Operator

A spectral reformulation of the Riemann hypothesis was obtained in [LaMa2] by the second author and H. Maier in terms of an inverse spectral problem for fractal strings. This problem is related to the question "Can one hear the shape of a fractal drum?" and was shown in [LaMa2] to have a positive answer for fractal strings whose dimension is c\in(0,1)-\{1/2} if and only if the Riemann hypothesis is true. Later on, the spectral operator was introduced heuristically by M. L. Lapidus and M. van Frankenhuijsen in their theory of complex fractal dimensions [La-vF2, La-vF3] as a map that sends the geometry of a fractal string onto its spectrum. We focus here on presenting the rigorous results obtained by the authors in [HerLa1] about the invertibility of the spectral operator. We show that given any $c\geq0$, the spectral operator $\mathfrak{a}=\mathfrak{a}_{c}$, now precisely defined as an unbounded normal operator acting in a Hilbert space $\mathbb{H}_{c}$, is `quasi-invertible' (i.e., its truncations are invertible) if and only if the Riemann zeta function $\zeta=\zeta(s)$ does not have any zeroes on the line $Re(s)=c$. It follows that the associated inverse spectral problem has a positive answer for all possible dimensions $c\in (0,1)$, other than the mid-fractal case when $c=1/2$, if and only if the Riemann hypothesis is true.Comment: To appear in: "Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics", Part 1 (D. Carfi, M. L. Lapidus, E. P. J. Pearse and M. van Frankenhuijsen, eds.), Contemporary Mathematics, Amer. Math. Soc., Providence, RI, 2013. arXiv admin note: substantial text overlap with arXiv:1203.482