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Motivated by studies in accelerator physics this paper computes theasymptotic 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 functionand $\displaystyle I_N\left(\frac{1}{k}\right)$ 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$\displaystyle \left[2,\left\lceil \frac{N}{k} \right\rceil\right]$ aredivisors of $N$.

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