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 ∫01(N(n,t)/ϕ(n)−t)2qdt, where n>1 and q>0 are positive integers, ϕ 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 ϕ(n) totatives of n relative to ϕ(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.
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BibTeX
@misc{brown2026calculation,
title = {{On the calculation of an integral introduced by Hubert Delange}},
author = {Steven Brown},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14249},
eprint = {2610.14249},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14249},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the calculation of an integral introduced by Hubert Delange
AU - Steven Brown
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.14249
UR - https://arxiv.org/abs/2610.14249
AB - 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 ∫_0^1(N(n,t)/ϕ(n)−t)^2qdt, where n>1 and q>0 are positive integers, ϕ 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 ϕ(n) totatives of n relative to ϕ(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.
ER -