A year ago I asked this question on Mathematics Stack Exchange with identifier 4245823 and same title Relevance of the deduction of similar theorems than Maier's theorem for other constellations of primes. I would like to know comments or answers for the question.
Wikipedia has the article Maier's theorem. I would like to ask as a soft question about the importance of the deduction of similar statements for (different) arithmetic functions counting the number of primes $p\leq x$ in different prime constellations (see the linked article). I know prime constellations as the primes in arithmetic progressions and different examples as the subsequences of prime numbers explained in pages 58-59 from the book [1]; that is the Exercise 1.35.
Question. I would like to know if you can glimpse what about the importance of study similar statements or variants of Maier's theorem for different sequences of prime numbers (constellations of prime numbers with infinitely many terms). Can these variants of Maier's theorem tell us any important thing about the distribution of prime numbers? Many thanks.
Previous Wikipedia's article also refers the issue about the original Maier's theorem and Cramér random model for prime numbers.
I hope that it is clear what I'm asking as a soft question; this is about what is the importance/relevance of attempting to prove new variants of Maier's theorem for constellations of primes (with a special importance in your opinion): if it is potentially interesting in the context of the study of the distribution of prime numbers (maybe, in the context of the open problems related to the distribution of prime numbers).
References:
[1] Richard Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective, Second Edition, Springer (2005).
