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

Abstract

Let $\omega^*(n)$ be the number of primes $p$ such that $p-1$ divides $n$. In 1955, Prachar proved that $\sum_{n\le x}\omega^*(n)^2=O\left(x(\log x)^2\right)$. Recently, Murty and Murty improved this to $$x(\log\log x)^3\ll\sum_{n\le x}\omega^*(n)^2\ll x\log x.$$ They further conjectured that there is some positive constant $C$ such that $$\sum_{n\le x}\omega^*(n)^2\sim Cx\log x$$ as $x\rightarrow \infty$. In a former note, the author gave the correct order of it by showing that $$\sum_{n\le x}\omega^*(n)^2\asymp x\log x.$$ In this subsequent article, we provide a conditional proof of their conjecture