The billiard in a regular polygon
Let \(P_ n\) be a regular \(n\)-gon. From the Ikehara Tauberian theorem one infers that there exists \(c_ n > 0\) such that the growth function of the length spectrum of the billiard in \(P_ n\) satisfies the asymptotic formula: \[ N(P_ n,t)\sim c_ n{t^ 2\over\| P_ n\|}\qquad (t\to \infty)\quad (\| P\|=\text{area}(P_ n)).\tag{1} \] The values \(c_ 3\) and \(c_ 4\) can be calculated directly from elementary considerations of lattice points associated to \(P_ n\). However, the value of \(c_ n\), \(n > 4\), depends of the prime factorization of \(n\). In order to exhibit this dependence, the author introduces a multiplicative number theoretic function \(c(n)\) (\(c(mn)=c(m)c(n)\) whenever \((m,n)=1\)), by giving its values for prime powers, \(n=p^ \nu\), \(\nu > 0\), as \[ c(n)=\begin{cases} 7n^ 3-6n^ 2 &\text{for \(n=2^ \nu\)}\\n^ 4\bigl(1+(1- n^{-2})/p(p+1)\bigr)&\text{for \(n=p^ \nu\), \(2<p\), \(p\) prime}.\end{cases}\tag{2} \] Also, let \(\varepsilon(n)=1\) or 2 as \(n\) is odd or even. The main purpose of the paper is to prove the Theorem: Let \(P_ n\) be a regular \(n\)-gon, \(n>4\). The asymptotic relation (1) is true with the constant \(c_ n\) given by \[ c_ n={c(n)- n^ 3\over 48\varepsilon (n)(n-2)\pi},\quad n>4,\quad\text{where}\quad c(n)\text{ is defined by (2)}. \]
scientific article; zbMATH DE number 93435