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

TL;DR

This paper proves a conjecture related to the sum of a function counting primes with specific divisibility properties, assuming a major unproven hypothesis, and introduces a novel sieve method for the proof.

Contribution

It confirms a conjecture by Murty and Murty on the asymptotic behavior of a prime-related sum under the Elliott--Halberstam conjecture, using a new sieve approach.

Findings

Confirmed the conjecture assuming Elliott--Halberstam conjecture

Developed a sieve method for prime sums in arithmetic progressions

Provided asymptotic formula for the sum involving sterisk(n)

Abstract

Let $\omega^*(n)$ 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 $$\sum_{n\leqslant x}\omega^*(n)^2\sim 2\frac{\zeta(2)\zeta(3)}{\zeta(6)}x\log x, \quad \text{as} \quad x\rightarrow \infty.$$ 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.