We prove the estimate Ek ≪ N/exp((1/4 + o(1)) log N log log log N/log log N), for the number Ek (N) of k-tuples (n + a 1, . . ., n + ak) of primes not exceeding N, for k of size c1 log N and N sufficiently large. A bound of this strength was previously known in the special case n - 2i (1 ≦ i <log n/log 2) only, (Vaughan, 1973). For general ai this is an improvement upon the work of Hofmann and Wolke (1996). The number of prime tuples of this size has considerable oscillations, when varying the prime pattern.
ASJC Scopus subject areas