## Last Updated:

2020-10-29T08:41:29.91+00:00

## DOI:

10.1007/s00222-019-00926-w

## abstract:

Let $\lambda$ denote the Liouville function. We show that, as $X \rightarrow

\infty$, $$\int_{X}^{2X} \sup_{\substack{P(Y)\in \mathbb{R}[Y]\\ deg(P)\leq k}}

\Big | \sum_{x \leq n \leq x + H} \lambda(n) e(-P(n)) \Big |\ dx = o ( X H)$$

for all fixed $k$ and $X^{\theta} \leq H \leq X$ with $0 < \theta < 1$ fixed

but arbitrarily small. Previously this was only established for $k \leq 1$. We

obtain this result as a special case of the corresponding statement for

(non-pretentious) $1$-bounded multiplicative functions that we prove. In fact,

we are able to replace the polynomial phases $e(-P(n))$ by degree $k$

nilsequences $\overline{F}(g(n) \Gamma)$. By the inverse theory for the Gowers

norms this implies the higher order asymptotic uniformity result

$$\int_{X}^{2X} \| \lambda \|_{U^{k+1}([x,x+H])}\ dx = o ( X )$$ in the same

range of $H$. We present applications of this result to patterns of various

types in the Liouville sequence. Firstly, we show that the number of sign

patterns of the Liouville function is superpolynomial, making progress on a

conjecture of Sarnak about the Liouville sequence having positive entropy.

Secondly, we obtain cancellation in averages of $\lambda$ over short polynomial

progressions $(n+P_1(m),\ldots, n+P_k(m))$, which in the case of linear

polynomials yields a new averaged version of Chowla's conjecture. We are in

fact able to prove our results on polynomial phases in the wider range $H\geq

\exp((\log X)^{5/8+\varepsilon})$, thus strengthening also previous work on the

Fourier uniformity of the Liouville function.

\infty$, $$\int_{X}^{2X} \sup_{\substack{P(Y)\in \mathbb{R}[Y]\\ deg(P)\leq k}}

\Big | \sum_{x \leq n \leq x + H} \lambda(n) e(-P(n)) \Big |\ dx = o ( X H)$$

for all fixed $k$ and $X^{\theta} \leq H \leq X$ with $0 < \theta < 1$ fixed

but arbitrarily small. Previously this was only established for $k \leq 1$. We

obtain this result as a special case of the corresponding statement for

(non-pretentious) $1$-bounded multiplicative functions that we prove. In fact,

we are able to replace the polynomial phases $e(-P(n))$ by degree $k$

nilsequences $\overline{F}(g(n) \Gamma)$. By the inverse theory for the Gowers

norms this implies the higher order asymptotic uniformity result

$$\int_{X}^{2X} \| \lambda \|_{U^{k+1}([x,x+H])}\ dx = o ( X )$$ in the same

range of $H$. We present applications of this result to patterns of various

types in the Liouville sequence. Firstly, we show that the number of sign

patterns of the Liouville function is superpolynomial, making progress on a

conjecture of Sarnak about the Liouville sequence having positive entropy.

Secondly, we obtain cancellation in averages of $\lambda$ over short polynomial

progressions $(n+P_1(m),\ldots, n+P_k(m))$, which in the case of linear

polynomials yields a new averaged version of Chowla's conjecture. We are in

fact able to prove our results on polynomial phases in the wider range $H\geq

\exp((\log X)^{5/8+\varepsilon})$, thus strengthening also previous work on the

Fourier uniformity of the Liouville function.

## Symplectic id:

1123722

## Download URL:

## Submitted to ORA:

Not Submitted

## Publication Type:

Journal Article