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

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