Solution to a problem of Luca, Menares and Pizarro-Madariaga
Volume 210 / 2023
Abstract
Let $k\ge 2$ be a positive integer and $P^+(n)$ the greatest prime factor of a positive integer $n$ with convention $P^+(1)=1$. For any $\theta \in \left [\frac 1{2k},\frac {17}{32k}\right )$, set $$T_{k,\theta }(x)=\sum _{\substack {p_1\cdot \cdot \cdot p_k\le x\\ P^+(\gcd (p_1-1,\ldots ,p_k-1))\ge (p_1\cdot \cdot \cdot p_k)^\theta }}1,$$ where the $p$’s are primes. It is proved that $$T_{k,\theta }(x)\ll _{k}\frac {x^{1-\theta (k-1)}}{(\log x)^2},$$ which, together with the lower bound $$T_{k,\theta }(x)\gg _{k}\frac {x^{1-\theta (k-1)}}{(\log x)^2}$$ obtained by Wu in 2019, answers a 2015 problem of Luca, Menares and Pizarro-Madariaga on the exact order of magnitude of $T_{k,\theta }(x)$.
A main novelty in the proof is that, instead of using the Brun–Titchmarsh theorem to estimate the $k$th moment of primes in arithmetic progressions, we give a transformation of this moment so that the task can be reduced to estimating certain clusters of primes.
