On a conjecture of M. R. Murty and V. K. Murty

Abstract Let $\omega ^*(n)$ be the number of primes p such that $p-1$ divides n. Recently, M. R. Murty and V. K. Murty proved that $$ \begin{align*}x(\log\log x)^3\ll\sum_{n\le x}\omega^*(n)^2\ll x\log x.\end{align*} $$ They further conjectured that there is some positive constant C such that $$ \begin{align*}\sum_{n\le x}\omega^*(n)^2\sim Cx\log x,\end{align*} $$ as $x\rightarrow \infty $ . In this short note, we give the correct order of the sum by showing that $$ \begin{align*}\sum_{n\le x}\omega^*(n)^2\asymp x\log x.\end{align*} $$