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

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

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