Integral Transforms of the Harmonic Sawtooth Map, The Riemann Zeta Function, Fractal Strings, and a Finite Reflection Formula

TL;DR

This paper explores the harmonic sawtooth map's fixed points, its Mellin transform's relation to the Riemann zeta function, and connections to fractal strings, Pythagorean triangles, and reflection formulas, revealing new analytical and geometric insights.

Contribution

It introduces a novel integral transform approach linking the harmonic sawtooth map to the Riemann zeta function and fractal geometry, including a finite approximation and reflection formula analysis.

Findings

Mellin transform of the harmonic sawtooth map analytically continues the zeta function.

The inverse scaling function's coefficients enumerate Large Schroder Numbers.

The fractal string's counting function matches Pythagorean triangle counts.

Abstract

The harmonic sawtooth map w(x) of the unit interval onto itself is defined where it is shown that its fixed points are enumerated by generating functions involving the golden ratio in their parameters. The appropriately scaled Mellin transform of w(x) is an analytic continuation of the Riemann zeta function {\zeta}(s) valid for all -Re(s) not an integer. The series expansion of the inverse scaling function which makes the Mellin transform of w(x) equal to the zeta function has coefficients enumerating the Large Schroder Numbers S_n, the number of perfect matchings in a triangular grid of n squares. A finite-sum approximation to is examined and an associated function is found which solves a reflection formula. The reflection function is singular at s = 0 and the residue at this point changes sign from negative to positive between the values of N = 176 and N = 177. The Gauss map h(x) is recalled so that its fixed points and Mellin transform can be contrasted to those of w(x). The geometric counting function of the fractal string associated to the lengths of the harmonic sawtooth map components happens to coincide with the counting function for the number of Pythagorean triangles of the form {(a,b,b+1):(b+1)<=x}. The volume of the inner tubular neighborhood of the boundary of the map with radius {\epsilon} is shown to have the particuarly simple closed-form. Also, the Minkowski content is shown to be 2 and the Minkowski dimension to be 1/2 and thus not invertible. Some definitions from the theory of fractal strings and membranes are also recalled.