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

2022/09/02 by Ding, Yuchen, Guo, Victor Zhenyu, Zhang, Yu
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2209.01087

Abstract

Let ω^*(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 ∑_{n\leqslant x}ω^*(n)²∼ 2(ζ(2)ζ(3))/(ζ(6))xlog x, as x→ ∞. 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.

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