On a Theorem of Bombieri-vinogradov Type

A Bombieri-vinogradov Theorem for All Number Fields

Abstract. The classical theorem of Bombieri and Vinogradov is generalized to a non-abelian, non-Galois setting. This leads to a prime number theorem of “mixed-type” for arithmetic progressions “twisted” by splitting conditions in number fields. One can view this as an extension of earlier work of M. R. Murty and V. K. Murty on a variant of the Bombieri-Vinogradov theorem. We develop this theory...

The Bombieri-vinogradov Theorem

A Bombieri-Vinogradov type exponential sum result with applications

We prove a Bombieri-Vinogradov type result for linear exponential sums over primes. Then we apply it to show that, for any irrational α and some θ > 0, there are infinitely many primes p such that p+ 2 has at most two prime factors and ‖αp+ β‖ < p−θ.

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 ...

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...