Explicit Formulas for Generalized Fractal Strings

Abstract

In this chapter, we obtain pointwise and distributional explicit formulas for the lengths and the frequencies of a fractal string. These explicit formulas express the counting function of the lengths or of the frequencies as a sum over the visible complex dimensions ω of the fractal string. To unify the exposition, and with a view toward later applications, we formulate our results in the language of generalized fractal strings, introduced in Chapter 4.