A Bombieri-vinogradov Theorem for All Number Fields

A Variant of the Bombieri-vinogradov Theorem in Short Intervals with Applications

We generalize the classical Bombieri-Vinogradov theorem to a short interval, non-abelian setting. This leads to variants of the prime number theorem for short intervals where the primes lie in arithmetic progressions that are “twisted” by a splitting condition in a Galois extension L/K of number fields. Using this result in conjunction with recent work of Maynard, we prove that rational primes ...

The Bombieri-vinogradov Theorem

The binary Goldbach problem with one prime of the form p = k 2 + l 2 + 1

We prove that almost all integers n ≡ 0 or 4 (mod 6) can be written in the form n = p1 + p2, where p1 = k 2 + l + 1 with (k, l) = 1. The proof is an application of the half-dimensional and linear sieves with arithmetic information coming from the circle method and the Bombieri-Vinogradov prime number theorem.

Small Gaps between Primes Ii (preliminary)

We examine an idea for approximating prime tuples. 1. Statement of results (Preliminary) In the present work we will prove the following result. Let pn denote the nth prime. Then (1.1) lim inf n→∞ (pn+1 − pn) log pn(log log pn)−1 log log log log pn < ∞. Further we show that supposing the validity of the Bombieri–Vinogradov theorem up to Q ≤ X with any level θ > 1/2 we have bounded differences b...

On a Theorem of Bombieri-vinogradov Type

for any e > 0 and A > 0, the implied constant in the symbol <g depending at most on E and A (see [1] and [14]). The original proofs of Bombieri and Vinogradov were greatly simplified by P. X. Gallagher [4]. An elegant proof has been given recently by R. C. Vaughan [13]. For other references see H. L. Montgomery [10] and H. -E. Richert [12]. Estimates of type (1) are required in various applicat...