On the calculation of an integral introduced by Hubert Delange

Abstract

In his paper, ''Sur la distribution des fractions irréductibles de dénominateur $n$ ou de dénominateur au plus égal à $x$'', Hubert Delange considers the family integrals defined by \begin{equation*} \int_0^1\left(\frac{N(n,t)}{ϕ(n)}-t\right)^{2q}dt, \end{equation*} where $n>1$ and $q>0$ are positive integers, $\varphi$ is the Euler totient function, and for any real number $t>0$, $N(n,t)$ counts the number reduced fractions whith denominator $n$ that belong to the interval $]0,t]$. He was able to evaluate the integral for $q=1$ but stated that ''Malheureusement, nous n'avons pas su jusqu'à présent faire cette évaluation pour $q>1$''. In addition to its intrinsic interest, this calculation would help understand the asymptotic distribution of the $\varphi (n)$ totatives of $n$ relative to $\varphi (n)$ equidistributed numbers in $[0,n]$. In this paper, we first provide some general properties of the integral for any $q>1$ and reduce the problem to its calculation for odd square-free integers with at least two distinct prime factors. We then analyse into more detail the case $q=2$ for which we provide a formula that reduces the problem to the evaluation of the fourfold Möbius sum, of integer valued modular sums related to generalized Dedekind sums.

arXiv comment15 pages