Fractal Strings and Number Theory:
The Harmonic String and the Prime String
Jason C. Payne, Michel L. Lapidus
Department of Mathematics
University of California, Riverside
A bstract
The purpose of this paper follows two veins which converge at one critical juncture- a bridge
between two seemingly disparate concepts. The first goal is to provide a brief survey of the topics
of fractal strings and their complex dimensions, which is achieved through the introduction of
their geometric and spectral zeta functions. Following this is consideration of two important
examples of generalized fractal strings- the harmonic string and the prime string. Through this
two discussions, the goal is to establish a strong connection between the concrete study of the
geometry and spectrum of fractal strings and the abstract world of number theory, which is
achieved by way of the infamous Riemann zeta function.
A U T H O R
Jason C. Payne
Pure Mathematics
Jason Payne is a graduating senior majoring in Pure Mathematics. His research
interests include fractal geometry and
complex analysis and their applications
in both analytic and algebraic number
theory, and Riemannian geometry. He is
F aculty M e n t o r
currently working on a senior thesis on
volume formulas in arbitrary (potentially
Michel L. Lapidus
Department of Mathematics
fractal) dimensions, and how they can
Michel Lapidus is Professor of Mathematics at UCR and also teaches in the
departments of Physics, Electrical Engineering and Computer Science. He works
at the crossroad of many research areas, including Mathematical Physics, Fractal
Geometry, Dynamical Systems, Parital Differential Equations, Noncommutative
Geometry, and Number Theory. His recent research books include “The Feynman Integral and
Feynman’s Operational Calculus” (Oxford Univ. Press, 2000, paperback: 2001; joint with G. W.
Jonson), “Fractal Geometry and Number Theory” (Birkhauser, 2000), “Fractal Geometry, Complex
Dimensions and Zeta Functions: Geometry and spectra of fractal strings” (Springer-Verlag, 2006)
[both joint with M. van Frankenhuysen], and most recently, “In Search of the Riemann Zeros: Strings,
fractal membranes and noncommutative spacetimes” (Amer. Math. Soc., Jan. 2008). Professor
Lapidus is a Fellow of the American Association for the Advancement of Science, and has been a
member of the American Mathematical Society Council since January 2002. Over the last nine years,
he has worked with nine undergraduate research projects and undergraduate Honors Theses.
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be combined with the study of fractal
strings in order to provide a generalization of Gauss’s Circle Problem. Some
fellow math majors and he are creating
of an official math club at UCR. After
graduating this spring, Jason will begin
the Mathematics Ph.D. program here at
UCR where he will continue his research
in fractal geometry, complex analysis,
and number theory.
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Fractal Strings and Number Theory: The Harmonic String and the Prime String
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Introduction
Introduction to Fractal Strings, and their Geometric and Spectral Zeta Functions
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Fractal Strings and Number Theory: The Harmonic String and the Prime String
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Fractal Strings and Number Theory: The Harmonic String and the Prime String
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Self-Similar Strings
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Generalized Fractal Strings
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References
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