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A U T H O R

Jason C. Payne

Pure Mathematics

Jason Payne is a graduating senior major-

ing 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

currently working on a senior thesis on

volume formulas in arbitrary (potentially

fractal) dimensions, and how they can

be combined with the study of fractal

strings in order to provide a generaliza-

tion 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.

Jason C. Payne, Michel L. LapidusDepartment of Mathematics University of California, Riverside

A B S T R A C T

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.

F A C U L T Y m e n T o R

Michel L. LapidusDepartment of MathematicsMichel 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.

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

Jason C. Payne

Introduction

Introduction to Fractal Strings, and their Geometric and Spectral Zeta Functions

U C R U n d e R g R a d U a t e R e s e a R C h J o U R n a l 3 7

FRACTAL STRINGS AND NUMBER THEORy: THE HARMONIC STRING AND THE PRIME STRING

Jason C. Payne

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FRACTAL STRINGS AND NUMBER THEORy: THE HARMONIC STRING AND THE PRIME STRING

Jason C. Payne

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FRACTAL STRINGS AND NUMBER THEORy: THE HARMONIC STRING AND THE PRIME STRING

Jason C. Payne

Self-Similar Strings

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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

Jason C. Payne

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FRACTAL STRINGS AND NUMBER THEORy: THE HARMONIC STRING AND THE PRIME STRING

Jason C. Payne

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FRACTAL STRINGS AND NUMBER THEORy: THE HARMONIC STRING AND THE PRIME STRING

Jason C. Payne

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FRACTAL STRINGS AND NUMBER THEORy: THE HARMONIC STRING AND THE PRIME STRING

Jason C. Payne

Generalized Fractal Strings

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FRACTAL STRINGS AND NUMBER THEORy: THE HARMONIC STRING AND THE PRIME STRING

Jason C. Payne

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FRACTAL STRINGS AND NUMBER THEORy: THE HARMONIC STRING AND THE PRIME STRING

Jason C. Payne

References