Discrete universality for Matsumoto zeta-functions and the nontrivial zeros of the Riemann zeta-function

In 2017, Garunk\v{s}tis, Laurin\v{c}ikas and Macaitien\.{e} proved the discrete universality theorem for the Riemann zeta-function sifted by the nontrivial zeros of the Riemann zeta-function. This discrete universality has been extended in various zeta-functions and $L$-functions. In this paper, we generalize this discrete universality for Matsumoto zeta-functions.Comment: 8 page

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

Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function

We survey some of the universality properties of the Riemann zeta function $\zeta(s)$ and then explain how to obtain a natural quantization of Voronin's universality theorem (and of its various extensions). Our work builds on the theory of complex fractal dimensions for fractal strings developed by the second author and M. van Frankenhuijsen in \cite{La-vF4}. It also makes an essential use of the functional analytic framework developed by the authors in \cite{HerLa1} for rigorously studying the spectral operator $\mathfrak{a}$ (mapping the geometry onto the spectrum of generalized fractal strings), and the associated infinitesimal shift $\partial$ of the real line: $\mathfrak{a}=\zeta(\partial)$. In the quantization (or operator-valued) version of the universality theorem for the Riemann zeta function $\zeta(s)$ proposed here, the role played by the complex variable $s$ in the classical universality theorem is now played by the family of `truncated infinitesimal shifts' introduced in \cite{HerLa1} to study the invertibility of the spectral operator in connection with a spectral reformulation of the Riemann hypothesis as an inverse spectral problem for fractal strings. This latter work provided an operator-theoretic version of the spectral reformulation obtained by the second author and H. Maier in \cite{LaMa2}. In the long term, our work (along with \cite{La5, La6}), is aimed in part at providing a natural quantization of various aspects of analytic number theory and arithmetic geometry

Remarks on the mixed joint universality for a class of zeta-functions

Two remarks related with the mixed joint universality for a polynomial Euler product and a periodic Hurwitz zeta-function with a transcendental parameter are given. One is the mixed joint functional independence, and the other is a generalized universality, which includes several periodic Hurwitz zeta-functions.Comment: 12 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

Riemann Zeroes and Phase Transitions via the Spectral Operator on Fractal Strings

The spectral operator was introduced by M. L. Lapidus and M. van Frankenhuijsen [La-vF3] in their reinterpretation of the earlier work of M. L. Lapidus and H. Maier [LaMa2] on inverse spectral problems and the Riemann hypothesis. In essence, it is a map that sends the geometry of a fractal string onto its spectrum. In this survey paper, we present the rigorous functional analytic framework given by the authors in [HerLa1] and within which to study the spectral operator. Furthermore, we also give a necessary and sufficient condition for the invertibility of the spectral operator (in the critical strip) and therefore obtain a new spectral and operator-theoretic reformulation of the Riemann hypothesis. More specifically, we show that the spectral operator is invertible (or equivalently, that zero does not belong to its spectrum) if and only if the Riemann zeta function zeta(s) does not have any zeroes on the vertical line Re(s)=c. Hence, it is not invertible in the mid-fractal case when c=1/2, and it is invertible everywhere else (i.e., for all c in(0,1) with c not equal to 1/2) if and only if the Riemann hypothesis is true. We also show the existence of four types of (mathematical) phase transitions occurring for the spectral operator at the critical fractal dimension c=1/2 and c=1 concerning the shape of the spectrum, its boundedness, its invertibility as well as its quasi-invertibility

Universality for mathematical and physical systems

All physical systems in equilibrium obey the laws of thermodynamics. In other words, whatever the precise nature of the interaction between the atoms and molecules at the microscopic level, at the macroscopic level, physical systems exhibit universal behavior in the sense that they are all governed by the same laws and formulae of thermodynamics. In this paper we describe some recent history of universality ideas in physics starting with Wigner's model for the scattering of neutrons off large nuclei and show how these ideas have led mathematicians to investigate universal behavior for a variety of mathematical systems. This is true not only for systems which have a physical origin, but also for systems which arise in a purely mathematical context such as the Riemann hypothesis, and a version of the card game solitaire called patience sorting.Comment: New version contains some additional explication of the problems considered in the text and additional reference