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AI InsightFinally, assuming the validity of a conjecture about the distribution of prime numbers along progressions with very large modulus, we demonstrate how it is possible to go beyond by showing that $s\leq (p-a)^{\epsilon}$ for every $\epsilon>0$ when $k=2$.
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Abstract
We obtain significant lower bounds for the number of shifted prime numbers having a relatively large prime power divisor, where being large has various quantifications. For any given $k\geq 2$, our results show the existence of infinitely many prime numbers $p$ that lie over certain admissible arithmetic progressions, and of the form $p=q^ks+a$ for suitable positive integers $a$, where $q$ is prime and $s$ is forced to be genuinely small with respect to $p$. We prove the existence of such prime numbers over progressions both unconditionally, and then conditionally by either assuming the nonexistence of Siegel zeros or weaker forms of the Riemann hypothesis for Dirichlet $L$-functions. Our approach allows us to provide considerable uniformity regarding the size of the modulus of the progressions, where the sought primes belong to, and the shift parameter $a$ by restricting the size of $s$ at the same time. Finally, assuming the validity of a conjecture about the distribution of prime numbers along progressions with very large modulus, we demonstrate how it is possible to go beyond by showing that $s\leq (p-a)^{\epsilon}$ for every $\epsilon>0$ when $k=2$.