Upper bounds for prime k -tuples of size log N and oscillations

Abstract

We prove the estimatefor the number E k (N ) of k-tuples (n + a 1 , . . . , n + a k ) of primes not exceeding N , for k of size c 1 log N and N sufficiently large. A bound of this strength was previously known in the special case n − 2 i (1 i < log n log 2 ) only, (Vaughan, 1973). For general a i 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.

Large prime gaps and probabilistic models

“…In the case of primes, it may be the case that (1.16) fails when |H| > log x log 2 x owing to potentially large fluctuations in both the size of S(H) and in the prime counts themselves. We note that Elsholtz [8] has shown that for any c > 0, the left side of (1.16) is bounded by…”

“…when |H| c log x, where the implied function o(1) depends on c. On the other hand, there are admissible tuples with |H| log x for which the left side of (1.16) is zero (see [8] for a construction of such H).…”

The exponential sum over squarefree integers

“…We now bound |Y a | from above using the following result by Elsholtz [2], which is based on the large sieve.…”