# Quantitative bounds for Gowers uniformity of the M\"obius and von Mangoldt functions

@inproceedings{Tao2021QuantitativeBF, title={Quantitative bounds for Gowers uniformity of the M\"obius and von Mangoldt functions}, author={Terence Tao and Joni Teravainen}, year={2021} }

We establish quantitative bounds on the U[N ] Gowers norms of the Möbius function μ and the von Mangoldt function Λ for all k, with error terms of shapeO((log logN)−c). As a consequence, we obtain quantitative bounds for the number of solutions to any linear system of equations of finite complexity in the primes, with the same shape of error terms. We also obtain the first quantitative bounds on the size of sets containing no k-term arithmetic progressions with shifted prime difference.

## One Citation

Discorrelation of multiplicative functions with nilsequences and its application on coefficients of automorphic $L$-functions

- Mathematics
- 2021

We introduce a class of multiplicative functions in which each function satisfies some statistic conditions, and then prove that above functions are not correlated with finite degree polynomial…

## References

SHOWING 1-10 OF 47 REFERENCES

COMPRESSIONS, CONVEX GEOMETRY AND THE FREIMAN–BILU THEOREM

- Mathematics
- 2005

We note a link between combinatorial results of Bollob\'as and Leader concerning sumsets in the grid, the Brunn-Minkowski theorem and a result of Freiman and Bilu concerning the structure of sets of…

Quantitative bounds in the inverse theorem for the Gowers $U^{s+1}$-norms over cyclic groups

- Mathematics
- 2018

We provide a new proof of the inverse theorem for the Gowers $U^{s+1}$-norm over groups $H=\mathbb Z/N\mathbb Z$ for $N$ prime. This proof gives reasonable quantitative bounds (the worst parameters…

AN INVERSE THEOREM FOR THE GOWERS U4-NORM

- MathematicsGlasgow Mathematical Journal
- 2010

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…

Siegel Zeros and Sarnak's Conjecture

- Mathematics
- 2021

Abstract. Assuming the existence of Siegel zeros, we prove that there exists an increasing sequence of positive integers for which Chowla’s Conjecture on k-point correlations of the Liouville…

Odd order cases of the logarithmically averaged Chowla conjecture

- Mathematics
- 2017

A famous conjecture of Chowla states that the Liouville function $\lambda(n)$ has negligible correlations with its shifts. Recently, the authors established a weak form of the logarithmically…

Multiple recurrence and convergence for sequences related to the prime numbers

- Mathematics
- 2006

For any measure preserving system (X, , μ,T) and A ∈ with μ(A) > 0, we show that there exist infinitely many primes p such that (the same holds with p − 1 replaced by p + 1). Furthermore, we show the…

Linear inequalities in primes

- MathematicsJournal d'Analyse Mathématique
- 2021

In this paper we prove an asymptotic formula for the number of solutions in prime numbers to systems of simultaneous linear inequalities with algebraic coefficients. For $m$ simultaneous inequalities…

On a theorem of Sárközy for difference sets and shifted primes

- Mathematics
- 2020

Abstract We show that if the difference of two elements of a set A ⊆ [ N ] is never one less than a prime number, then | A | = O ( N exp ( − c ( log N ) 1 / 3 ) ) for some absolute constant c > 0…

A new proof of Szemerédi's theorem

- Mathematics
- 2001

In 1927 van der Waerden published his celebrated theorem on arithmetic progressions, which states that if the positive integers are partitioned into finitely many classes, then at least one of these…

Opera De Cribro

- Computer Science
- 2010

A wide range of applications are included, both to traditional questions such as those concerning primes, and to areas previously unexplored by sieve methods, such as elliptic curves, points on cubic surfaces and quantum ergodicity.