Multiplicative functions in short intervals

By Kaisa Matomaki, Maksym Radziwill. 2015

182 citations recorded by the source corpus.

We introduce a general result relating "short averages" of a multiplicative function to "long averages" which are well understood. This result has several consequences. First, for the M\"obius function we show that there are cancellations in the sum of $\mu(n)$ in almost all intervals of the form $[x, x + \psi(x)]$ with $\psi(x) \rightarrow \infty$ arbitrarily slowly. This goes beyond what was previously known conditionally on the Density Hypothesis or the stronger Riemann Hypothesis. Second, we settle the long-standing conjecture on the existence of $x^{\epsilon}$-smooth numbers in intervals of the form $[x, x + c(\varepsilon) \sqrt{x}]$, recovering unconditionally a conditional (on the Riemann Hypothesis) result of Soundararajan. Third, we show that the mean-value of $\lambda(n)\lambda(n+1)$, with $\lambda(n)$ Liouville's function, is non-trivially bounded in absolute value by $1 - \delta$ for some $\delta > 0$. This settles an old folklore conjecture and constitutes progress towards Chowla's conjecture. Fourth, we show that a (general) real-valued multiplicative function $f$ has a positive proportion of sign changes if and only if $f$ is negative on at least one integer and non-zero on a positive proportion of the integers. This improves on many previous works, and is new already in the case of the M\"obius function. We also obtain some additional results on smooth numbers in almost all intervals, and sign changes of multiplicative functions in all intervals of square-root length.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dd0163aacdb4f2109127e9be66d1e7e9d503a70b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

Sarnak’s Conjecture: What’s New (2017, background): “One more consequence of the main result in [124] is the following: Theorem 2.”
THE LOGARITHMICALLY AVERAGED CHOWLA AND ELLIOTT CONJECTURES FOR TWO-POINT CORRELATIONS (2015, methodology): “In the recent paper [21] it was shown that in fact all four sign patterns occur x times, so in particular ∣∣∣∣∑ n6x λ(n)λ(n + 1) ∣∣∣∣ 6 (1− δ)x”
The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures (2017, methodology): “8] (see also [24], [32]), we conclude that the length five sign patterns ++++−, −−−−+, −++++, and +−−−− each occur for the Liouville function with positive lower logarithmic density.”
The logarithmic Sarnak conjecture for ergodic weights (2017, methodology): “3 use, in addition to the previous number theoretic tools, a recent result of Tao [65] on the twopoint correlations of the Liouville function which in turn depends upon a recent result of Matomäki and Radziwiłł [53] on averages of the Möbius and the Liouville function on short intervals.”