The difference between consecutive primes in an arithmetic progression

Abstract

We obtain new results concerning the simultaneous distribution of prime numbers in arithmetic progressions and in short intervals. For example, we show that there is an absolute constant δ > 0 such that 'almost all' arithmetic progressions a mod q with q ≤ x δ and (a, q) = 1 contain prime numbers from the interval (x − x 0.53 , x].

On the Number of Carmichael Numbers up to x

“…We will sketch the proof of Theorem 2 by showing that the arithmetical information available here is exactly analogous to that in [11] and [16]. It follows that we obtain the corresponding exponent (compare the similar argument in [14]). …”

“…The author's sieve method (described in [9] and [10], and developed in [3,4,11,14]), shows how this can be done using Buchstab's identity and asymptotic formulae for sums of the form…”

Bounded gaps between primes in short intervals

“…In this section we outline a proof of Theorem 2.3, the extension of Kumchev's main result in [12] to all exponents δ ≥ 0.525. To do this, we will synthesize the argument of Kumchev in [12], which shows the result with 0.53 in place of 0.525, and the argument of Baker, Harman, and Pintz in [3].…”

“…In this section we outline a proof of Theorem 2.3, the extension of Kumchev's main result in [12] to all exponents δ ≥ 0.525. To do this, we will synthesize the argument of Kumchev in [12], which shows the result with 0.53 in place of 0.525, and the argument of Baker, Harman, and Pintz in [3]. To modify the results of [3] for use with primes in arithmetic progressions, we replace the use of Watt's theorem by its extension to Dirichlet L-functions in [7].…”

“…Remark 2.4. We can take the implied constant in (ii) to be ≥ 1 − β for some β < 1 made explicit in the computations of [12] and [3]. For sufficiently small ǫ, we can take β ≤ 0.94.…”