Asymptotic behavior of a series of Euler's totient function $\phi (k)$ 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 $\sum _{k=1}^N\phi (k)I_N\left(\frac{1}{k}\right)$, where $\phi (k)$ is Euler's Totient function and $I_N\left(\frac{1}{k}\right)$ is the position that $1/k$ occupies in the Farey sequence of order $N$. To this end an exact formula for $I_N\left(\frac{1}{k}\right)$ is derived when all integers in $\left[2,\lceil \frac{N}{k}\rceil \right]$ are divisors of $N$.

6 pages