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

TL;DR

This paper analyzes the asymptotic behavior of a series involving Euler's totient function and the position of fractions in Farey sequences, providing an exact formula under specific divisor conditions.

Contribution

It derives an exact formula for the position of 1/k in Farey sequences when certain divisor conditions are met, advancing understanding of Farey sequence properties.

Findings

Asymptotic behavior characterized for the series involving $\

Exact formula for $I_N(1/k)$ under divisor conditions derived.

Insights into Farey sequence structure and totient function interactions.

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 $N$.