Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Integral Transforms of the Harmonic Sawtooth Map, The Riemann Zeta
  Function, Fractal Strings, and a Finite Reflection Formula   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.

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The categories of this paper are: solv-int
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The categories of this paper are: ao-sci
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The categories of this paper are: cmp-lg
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The categories of this paper are: astro-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: eess
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The categories of this paper are: math-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: physics
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The categories of this paper are: funct-an
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The categories of this paper are: plasm-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-ph