A criterion for nilpotency of a finite group by the sum of element orders

Denote the sum of element orders in a finite group $G$ by $\psi (G)$ and let $C_n$ denote the cyclic group of order $n$. In this paper, we prove that if $\mathrm{\mid }G\mathrm{\mid }=n$ and $\psi (G)>\frac{13}{21}\psi (C_n)$, then $G$ is nilpotent. Moreover, we have $\psi (G)=\frac{13}{21}\psi (C_n)$ if and only if $n=6m$ with $(6,m)=1$ and $G\cong S_3\times C_m$. Two interesting consequences of this result are also presented.