Asymptotic behavior of a series of Euler's totient function $\varphi(k)$ times the index of $1/k$ in a Farey sequence

Abstract

Motivated by studies in accelerator physics this paper computes the asymptotic behavior of the series $\displaystyle \sum_{k=1}^N \varphi(k) I_N\left(\frac{1}{k}\right)$, where $\varphi(k)$ is Euler's Totient function and $\displaystyle I_N\left(\frac{1}{k}\right)$ is the position that $1/k$ occupies in the Farey sequence of order $N$. To this end an exact formula for $\displaystyle I_N\left(\frac{1}{k}\right)$ is derived when all integers in $\displaystyle \left[2,\left\lceil \frac{N}{k} \right\rceil\right]$ are divisors of $N$.