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

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

Fractal Strings and Multifractal Zeta Functions

For a Borel measure on the unit interval and a sequence of scales that tend to zero, we define a one-parameter family of zeta functions called multifractal zeta functions. These functions are a first attempt to associate a zeta function to certain multifractal measures. However, we primarily show that they associate a new zeta function, the topological zeta function, to a fractal string in order to take into account the topology of its fractal boundary. This expands upon the geometric information garnered by the traditional geometric zeta function of a fractal string in the theory of complex dimensions. In particular, one can distinguish between a fractal string whose boundary is the classical Cantor set, and one whose boundary has a single limit point but has the same sequence of lengths as the complement of the Cantor set. Later work will address related, but somewhat different, approaches to multifractals themselves, via zeta functions, partly motivated by the present paper.Comment: 32 pages, 9 figures. This revised version contains new sections and figures illustrating the main results of this paper and recent results from others. Sections 0, 2, and 6 have been significantly rewritte

Ihara zeta functions for periodic simple graphs

The definition and main properties of the Ihara zeta function for graphs are reviewed, focusing mainly on the case of periodic simple graphs. Moreover, we give a new proof of the associated determinant formula, based on the treatment developed by Stark and Terras for finite graphs.Comment: 17 pages, 7 figures. V3: minor correction