Sharp order of magnitude in the hidden-transition regime

Determine the sharp order of magnitude of H^*(x,y,z) in the regime where \gamma\to0 and \lambda_0\to\infty, thereby resolving whether the factor V in the bounds of Theorem 1(iv) reflects a genuine clustering phenomenon or merely a limitation of the proof method.

Background

Theorem 1 determines the order of magnitude of H*(x,y,z) in all stated parameter ranges except when the auxiliary factor V tends to infinity in part (iv). In the simultaneous regime \gamma\to0 and \lambda_0\to\infty, the authors obtain matching estimates only up to a potentially nontrivial factor.

They explicitly question whether this loss arises from the method or from clustering of the logarithms of shifted-prime divisors, analogous to the clustering phenomenon in the generic divisor problem. The conjectured formula below gives the proposed sharp order.

References

One may ask whether this failure to produce the correct order is the failure of the method, or the effect of clustering as in thm:FORDANNALS0. We are inclined to believe the latter meaning that neither our upper nor our lower bound are sharp in this case. We conjecture that the true order of magnitude is as follows.

thm:FORDANNALS0:

H(x,y,z)≍xuδlog⁡(2/u)−3/2 ,H (x,y,z)\asymp xu^{\delta}\log (2/u)^{-3/2}\,,

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

We conjecture that the true order of magnitude is as follows.

\begin{conj} Let $\lambda_0 = \log\log z\prime-\log\log (z/y)$. Then for $\gamma \le 0.01$ we have

x{-1}H*(x,y,z) \asymp (\log z\prime){-Q(\frac{1+\gamma}{\log 2})} (\log\log z\prime){-1/2} \cdot (\gamma + (\lambda_0+1){-1/2})\,.

\end{conj}

— Integers divisible by a shifted prime in a given interval  (2608.27356 - Abdallah et al., 27 Aug 2026) in Conjecture following the discussion of Theorem 1(iv), Section 1