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

TL;DR

This paper investigates the asymptotic behavior of the sum of the squares of the number of primes p where p-1 divides n, confirming it grows proportionally to x log x.

Contribution

It establishes the correct order of the sum, showing it is asymptotically proportional to x log x, refining previous bounds and conjecture.

Findings

Sum of ω*(n)^2 is asymptotically proportional to x log x.

Provides the precise order of growth confirming the conjecture.

Refines previous bounds by Murty and Murty.

Abstract

Let $\omega^*(n)$ be the number of primes $p$ such that $p-1$ divides $n$. Recently, R. M. Murty and V. K. Murty proved that $$x(\log\log x)^3\ll\sum_{n\le x}\omega^*(n)^2\ll x\log x.$$ 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 this short note, we give the correct order of the sum by showing that $$\sum_{n\le x}\omega^*(n)^2\asymp x\log x.$$