Fractal Strings and Multifractal Zeta Functions

Abstract

Abstract. 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.

Towards quantized number theory: spectral operators and an asymmetric criterion for the Riemann hypothesis

“…We note that recent developments in the theory are described in [7, ch. 13], including a first attempt at a higher dimensional theory of complex dimensions for the special case of fractal sprays (in the sense of [30]) and self-similar tilings (see [7, §13.1], based on [63][64][65][66]72]), p-adic fractal strings and associated non-Archimedean fractal tube formulae (see [7, §13.2], based on [56][57][58][59][60]), multi-fractal zeta functions and their 'tapestries' of complex dimensions (see [7, §13.3], based on [50,55,67]), random fractal strings (such as stochastically self-similar strings and the zero-set of Brownian motion) and their spectra (see [7, §13.4], based on [51]), as well as a new approach to the RH based on a conjectural Riemann flow of fractal membranes (i.e. quantized fractal strings) and correspondingly flows of zeta functions (or 'partition functions') and of the associated zeros (see [7, §13.5], which gives a brief overview of the aforementioned book [20], In search of the Riemann zeros).…”

The Sound of Fractal Strings and the Riemann Hypothesis

“…For general references concerning the theory of the Riemann zeta function and related aspects of analytic number theory, we mention, for example, [Edw,Ing,Ivi,Lap6,Pat,Ser,Ti] along with the relevant references therein. For fractal string theory and the associated theory of complex dimensions, along with their applications to a variety of subjects, including fractal geometry, spectral geometry, number theory and dynamical systems, we refer to , along with [EllLapMaRo,HamLap,Fal2,HeLap,LapLéRo,, LapLu-vFr1-2, LapMai1-2, LapNe, LapPe1-3, LapPeWi1-2, LapRoZu,LéMen,MorSepVi,Pe,PeWi,Ra,RatWi2, and the relevant references therein. In particular, Chapter 13 of provides an exposition of a number of recent extensions and applications of the theory, including to fractal sprays (higher-dimensional analogs of fractal strings, [LapPom3]) and self-similar systems §13.1], based on [LapPe2-3, LapPeWi1-2, Pe, PeWi]), p-adic (or nonarchimedean) geometry, §13.2], based on ), multifractals §13.3], based on [LapRo, LapLéRo, EllLapMaRo]), random fractal strings §13.4], based on [HamLap]), as well as fractal membranes and the Riemann (or modular) flow on the moduli space of fractal membranes §13.5], based on the book [Lap6] and on [LapNe]).…”

Local complex dimensions of a fractal string

“…The theory of fractal strings has been generalized in various directions: a random setting is considered in [5]; higher-dimensional extensions are proposed in [13,16]; and multifractal zeta functions are investigated in [11,15,16].…”