Asymptotic formula for the shifted-prime divisor counting function

Establish whether an asymptotic formula of the form H^*(x,y,z)=x\rho(y,z)(1+o_{x\to\infty}(1)) exists for all admissible pairs y,z, where H^*(x,y,z) counts integers n\le x divisible by some shifted prime p-1 with p\in(y,z].

Background

The paper studies H*(x,y,z), the number of integers up to x divisible by at least one shifted prime p-1 with p in (y,z]. Its main theorem determines the order of magnitude of this quantity for almost all intervals and identifies several phase transitions according to the interval parameters.

The authors note that a full asymptotic is known in some regimes, but that the existence of an asymptotic proportion uniformly for all y and z remains unresolved. This is presented as a broader difficult problem related to the corresponding question for generic divisors.

References

Whether an asymptotic $H(x,y,z) = x \rho(y,z) (1+ o_{x \to \infty}(1))$ exists for all $y,z$ remains a difficult open question, however, in their recent work, Green and Sawhney showed that the permutation analog of $H(x,y,2y)$ indeed possesses an asymptotic when $y = x{o(1)}$.

— Integers divisible by a shifted prime in a given interval  (2608.27356 - Abdallah et al., 27 Aug 2026) in Section 1, Introduction