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

Let ${\omega }^{\ast }(n)$ be the number of primes $p$ such that $p-1$ divides $n$. Assuming the Elliott–Halberstam Conjecture, we prove a conjecture posted by M. R. Murty and V. K. Murty in 2021 which states that $$\sum _{n⩽x}{\omega }^{\ast }(n{)}^2\sim 2\frac{\zeta (2)\zeta (3)}{\zeta (6)}x\log x,\text{as}x\to \mathrm{\infty }\mathrm{.}$$ The above sum was first investigated by Prachar in 1955. One of the key ingredients in our argument is the application of a sieve result on estimating various certain summations involving primes in arithmetic progressions, rather than a direct use of the Brun–Titchmarsh inequality which would not be applicable for our task.