On the difference between consecutive primes

The distribution of prime ideals of imaginary quadratic fields

“…• the exponent 7/12 + ε obtained by Coleman [4, Theorem 1.1] corresponds to a celebrated result by Huxley [13] (henceforth, ε denotes a sufficiently small, positive constant, not necessarily the same in each occurrence); • the exponent 11/20 that follows from [7, Theorem 4] corresponds to the main result of Heath-Brown and Iwaniec [12]; • Theorem 1 corresponds to the existence of rational primes in all intervals (x − x 0.53 , x] with x ≥ x 0 , and compares with recent results by Baker and Harman [1] and by Baker, Harman and Pintz [3] in which the lengths of the intervals are x 0.535 and x 0.525 , respectively. Our method is an adaptation to algebraic number fields of the sieve method introduced by the first author [8,9] and shares many features with [2] and [3].…”

On the fractional parts of $x/n$ and related sequences. II

“…his with the choice 8 == exp ( --Cg ( ---?--) ) and 'V 2 \loglog^/ / Huxley's theorem [2] that (6.2) holds with C = 12/5 establishes Theorem 10.…”

“…-Clearly Ja-v/2 P P t/CT/2 P P 'a-v/2 | p -»2<a;u Thus, the double sum on the right of (6.9) converges absolutely, and uniformly in v on [1,2]. Thus, by (6.6), (6.8), (6, 9) and (6.10) If the Riemann hypothesis is assumed, then at once from (6.11) and (6.13), 2i<^ 5 621ogY+^ S ^^ ^flog-J-Y.…”

“…By (2), page 226, of Walfisz [II], we have (6. The sum £3 is estimated in the same way. By f6.16), f6 11) (6.17) and (6.2), ' "…”

A Note on Primes Between Consecutive Powers

“…This paper concerns with the distribution of prime numbers between two consecutive powers of integers, as a natural generalization of the above problem. The well known result of M. N. Huxley [8] about the distribution of prime in short intervals implies that all intervals [n a ; (n 1) a ] contain the expected number of primes for a > 12 5 and n 3 I. This was slightly improved by D. R. HeathBrown [7] to a !…”