**Memoirs of the American Mathematical Society**

1997;
97 pp;
Softcover

MSC: Primary 35; 11; 28; 34; 58;
Secondary 26; 47; 78

Print ISBN: 978-0-8218-0597-8

Product Code: MEMO/127/608

List Price: $46.00

AMS Member Price: $27.60

MAA member Price: $41.40

**Electronic ISBN: 978-1-4704-0193-1
Product Code: MEMO/127/608.E**

List Price: $46.00

AMS Member Price: $27.60

MAA member Price: $41.40

# Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings and the Riemann Zeta-Functions

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*Christina Q. He; Michel L. Lapidus*

This memoir provides a detailed study of the effect of non power-like irregularities of (the geometry of) the fractal boundary on the spectrum of “fractal drums” (and especially of “fractal strings”).

In this work, the authors extend previous results in this area by using the notion of generalized Minkowski content which is defined through some suitable “gauge functions” other than power functions. (This content is used to measure the irregularity (or “fractality”) of the boundary of an open set in \(R^n\) by evaluating the volume of its small tubular neighborhoods.) In the situation when the power function is not the natural “gauge function”, this enables the authors to obtain more precise estimates, with a broader potential range of applications than in previous papers of the second author and his collaborators.

#### Readership

Graduate students and research mathematicians interested in dynamical systems, fractal geometry, partial differential equations, analysis, measure theory, number theory or spectral geometry. Physicists interested in fractal geometry, condensed matter physics or wave propagation in random or fractal media.

#### Table of Contents

# Table of Contents

## Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings and the Riemann Zeta-Functions

- Contents vii8 free
- 1 Introduction 112 free
- 2 Statement of the Main Results 617 free
- 3 Sharp Error Estimates and their Converse when n = 1 1425
- 4 Spectra of Fractal Strings and the Riemann Zeta-Function 3849
- 5 The Complex Zeros of the Riemann Zeta-Function 5768
- 6 Error Estimates for n ≥ 2 6980
- 7 Examples 8091
- Appendix: Examples of Gauge Functions 8899
- References 94105