Number Theory


Integral Transforms of the Harmonic Sawtooth Map, The Riemann Zeta Function, and Fractal Strings

Authors: Stephen Crowley

The harmonic sawtooth map w(x) of the unit interval onto itself is defined. It is shown that its fixed points {x: w(x) = x} are enumerated by the n-th derivatives of a Meijer-G function and Lerch transcendent, serving as exponential and ordinary generating functions respectively, and involving the golden ratio in their parameters. The appropriately scaled Mellin transform of w(x) is an analytic continuation of the Riemann zeta function ζ(s) valid ∀−Re(s) not in N. 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 Schröder Numbers S_n, defined as the number of perfect matchings in a triangular grid of n squares and expressible as a hypergeometric function. A finite-sum approximation to ζ(s) denoted by ζw(N;s) is examined and an associated function χ(N;s) is found which solves the reflection formula ζw(N;1−s)=χ(N;s)ζw(N;s). The function χ(N;s) 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. Some rather elegant graphs of the reflection functions χ(N;s) are also provided. The values ζw(1-N;s) are found to be related to the Bernoulli numbers. The Mellin and Laplace transforms of the individual component functions of the infinite sums and their roots are compared. 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 NLw(x) = floor(sqrt(2x+1)/2-1/2) of the fractal string Lw associated to the lengths of the harmonic sawtooth map components {w_n(x)}n=1..∞ 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 ∂Lw with radius ε is shown to have the particuarly simple closed-form VLw(ε) = (4εv(ε)^2−4εv(ε)+1)/2v(ε) where v(ε) = floor((ε+sqrt(ε^2+ε))/2ε). Also, the Minkowski content of Lw is shown to be MLw = 2 and the Minkowski dimension to be DLw = 1/2 and thus not invertible. The geometric zeta function, which is the Mellin transform of the geometric counting function NLw(x), is calculated and shown to have a rather unusual closed form involving a finite sum of Riemann zeta functions and binomioal coefficients. Some definitions from the theory of fractal strings and membranes are also recalled.

Comments: 39 Pages.

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Submission history

[v1] 2012-02-26 22:36:06
[v2] 2012-03-01 21:00:49
[v3] 2012-03-30 14:42:21
[v4] 2012-03-30 18:13:20
[v5] 2012-04-02 14:52:25
[v6] 2012-04-15 11:27:21
[v7] 2012-04-20 16:56:04
[v8] 2012-05-04 15:57:56
[v9] 2012-05-10 21:59:51
[vA] 2012-05-15 14:18:48
[vB] 2012-05-18 13:09:21

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