Studying the relationship between the geometry, arithmetic and
spectra of fractals has been a subject of significant interest in
contemporary mathematics. This book contributes to the literature
on the subject in several different and new ways. In particular,
the authors provide a rigorous and detailed study of the spectral
operator, a map that sends the geometry of fractal strings onto
their spectrum. To that effect, they use and develop methods from
fractal geometry, functional analysis, complex analysis, operator
theory, partial differential equations, analytic number theory and
mathematical physics.Originally, M L Lapidus and M van
Frankenhuijsen 'heuristically' introduced the spectral operator in
their development of the theory of fractal strings and their
complex dimensions, specifically in their reinterpretation of the
earlier work of M L Lapidus and H Maier on inverse spectral
problems for fractal strings and the Riemann hypothesis.One of the
main themes of the book is to provide a rigorous framework within
which the corresponding question 'Can one hear the shape of a
fractal string?' or, equivalently, 'Can one obtain information
about the geometry of a fractal string, given its spectrum?' can be
further reformulated in terms of the invertibility or the
quasi-invertibility of the spectral operator.The infinitesimal
shift of the real line is first precisely defined as a
differentiation operator on a family of suitably weighted Hilbert
spaces of functions on the real line and indexed by a dimensional
parameter c. Then, the spectral operator is defined via the
functional calculus as a function of the infinitesimal shift. In
this manner, it is viewed as a natural 'quantum' analog of the
Riemann zeta function. More precisely, within this framework, the
spectral operator is defined as the composite map of the Riemann
zeta function with the infinitesimal shift, viewed as an unbounded
normal operator acting on the above Hilbert space.It is shown that
the quasi-invertibility of the spectral operator is intimately
connected to the existence of critical zeros of the Riemann zeta
function, leading to a new spectral and operator-theoretic
reformulation of the Riemann hypothesis. Accordingly, the spectral
operator is quasi-invertible for all values of the dimensional
parameter c in the critical interval (0,1) (other than in the
midfractal case when c =1/2) if and only if the Riemann hypothesis
(RH) is true. A related, but seemingly quite different,
reformulation of RH, due to the second author and referred to as an
'asymmetric criterion for RH', is also discussed in some detail:
namely, the spectral operator is invertible for all values of c in
the left-critical interval (0,1/2) if and only if RH is true.These
spectral reformulations of RH also led to the discovery of several
'mathematical phase transitions' in this context, for the shape of
the spectrum, the invertibility, the boundedness or the
unboundedness of the spectral operator, and occurring either in the
midfractal case or in the most fractal case when the underlying
fractal dimension is equal to 1/2 or 1, respectively. In
particular, the midfractal dimension c=1/2 is playing the role of a
critical parameter in quantum statistical physics and the theory of
phase transitions and critical phenomena.Furthermore, the authors
provide a 'quantum analog' of Voronin's classical theorem about the
universality of the Riemann zeta function. Moreover, they obtain
and study quantized counterparts of the Dirichlet series and of the
Euler product for the Riemann zeta function, which are shown to
converge (in a suitable sense) even inside the critical strip.For
pedagogical reasons, most of the book is devoted to the study of
the quantized Riemann zeta function. However, the results obtained
in this monograph are expected to lead to a quantization of most
classic arithmetic zeta functions, hence, further 'naturally
quantizing' various aspects of analytic number theory and
arithmetic geometry.The book should be accessible to experts and
non-experts alike, including mathematics and physics graduate
students and postdoctoral researchers, interested in fractal
geometry, number theory, operator theory and functional analysis,
differential equations, complex analysis, spectral theory, as well
as mathematical and theoretical physics. Whenever necessary,
suitable background about the different subjects involved is
provided and the new work is placed in its proper historical
context. Several appendices supplementing the main text are also
included.
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