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Assuming a variant of the Hardy--Littlewood prime k-tuple conjecture, we compute mixed moments of the number of primes in disjoint short intervals and short progressions respectively. This involves estimating the mean of singular series over any product of lattices, which is of independent interest. As a consequence, we establish the convergence of both sequences of suitably normalized primes to a standard Poisson point process.
Sun‐Kai Leung (Sun,) studied this question.