The Bombieri-Vinogradov theorem for nilsequences

@article{Shao2020TheBT,
title={The Bombieri-Vinogradov theorem for nilsequences},
author={Xuancheng Shao and Joni Teravainen},
journal={arXiv: Number Theory},
year={2020}
}
• Published 10 June 2020
• Mathematics
• arXiv: Number Theory
We establish results of Bombieri-Vinogradov type for the von Mangoldt function $\Lambda(n)$ twisted by a nilsequence. In particular, we obtain Bombieri-Vinogradov type results for the von Mangoldt function twisted by any polynomial phase $e(P(n))$; the results obtained are as strong as the ones previously known in the case of linear exponential twists. We derive a number of applications of these results. Firstly, we show that the primes $p$ obeying a "nil-Bohr set" condition, such as $\|\alpha… 2 Citations • Mathematics, Computer Science • 2021 A transference principle which applies to general affine-linear configurations of finite complexity and shows that in these sets of primes the existence of solutions to finite complexity systems of linear equations is determined by natural local conditions. We survey techniques used to detect prime numbers in sets, highlighting the strengths and limitations of current techniques. References SHOWING 1-10 OF 45 REFERENCES We prove Bombieri--Vinogradov and Barban--Davenport--Halberstam type theorems for the y-smooth numbers less than x, on the range log^{K}x \leq y \leq x. This improves on the range \exp{log^{2/3 + The Green-Tao-Ziegler theorem provides asymptotics for the number of prime tuples of the form$(\psi_1(n),\ldots,\psi_t(n))$when$n$ranges among the integer vectors of a convex body$K\subset
• Mathematics
Journal of the European Mathematical Society
• 2022
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• 2007
A theorem of Leibman asserts that a polynomial orbit $(g(1),g(2),g(3),\ldots)$ on a nilmanifold $G/\Gamma$ is always equidistributed in a union of closed sub-nilmanifolds of $G/\Gamma$. In this paper