The Bombieri-Vinogradov theorem for nilsequences

  title={The Bombieri-Vinogradov theorem for nilsequences},
  author={Xuancheng Shao and Joni Teravainen},
  journal={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… 

A transference principle for systems of linear equations, and applications to almost twin primes

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.

Counting primes

We survey techniques used to detect prime numbers in sets, highlighting the strengths and limitations of current techniques.



Bombieri-Vinogradov for multiplicative functions, and beyond the x1/2-barrier

Bombieri--Vinogradov and Barban--Davenport--Halberstam type theorems for smooth numbers

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 +

A higher-dimensional Siegel-Walfisz theorem

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

On the Möbius function in all short intervals

We show that, for the Mobius function $\mu(n)$, we have $$ \sum_{x 0.55$. This improves on a result of Ramachandra from 1976, which is valid for $\theta>7/12$. Ramachandra's result corresponded to

Vinogradov’s three primes theorem with almost twin primes

In this paper we prove two results concerning Vinogradov’s three primes theorem with primes that can be called almost twin primes. First, for any $m$ , every sufficiently large odd integer $N$ can be

Bounded gaps between primes in Chebotarev sets

AbstractPurposeA new and exciting breakthrough due to Maynard establishes that there exist infinitely many pairs of distinct primes p1, p2 with |p1-p2| ≤ 600 as a consequence of the

On exponential sums over primes in arithmetic progressions

with a wide uniformity in real a, where A is the von Mangoldt function and, for real 6, e{6) = exp(2niO). By using a combinatrial identity, R. C. Vaughan presented an elegant simple argument on it,


Abstract We prove the so-called inverse conjecture for the Gowers Us+1-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we show that if f : [N] → ℂ is a

Bounded gaps between primes in short intervals

Baker, Harman, and Pintz showed that a weak form of the Prime Number Theorem holds in intervals of the form $$[x-x^{0.525},x]$$[x-x0.525,x] for large x. In this paper, we extend a result of Maynard

The quantitative behaviour of polynomial orbits on nilmanifolds

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