A1 Refereed original research article in a scientific journal
Multiplicative functions in short intervals
Authors: Matomaki K, Radziwill M
Publisher: ANNAL MATHEMATICS
Publication year: 2016
Journal: Annals of Mathematics
Journal name in source: ANNALS OF MATHEMATICS
Journal acronym: ANN MATH
Volume: 183
Issue: 3
First page : 1015
Last page: 1056
Number of pages: 42
ISSN: 0003-486X
DOI: https://doi.org/10.4007/annals.2016.183.3.6
Abstract
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 Mobius 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) -> infinity 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(epsilon)root 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 nontrivially bounded in absolute value by 1 - delta (5 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 nonzero on a positive proportion of the integers. This improves on many previous works and is new already in the case of the Mobius 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.
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 Mobius 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) -> infinity 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(epsilon)root 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 nontrivially bounded in absolute value by 1 - delta (5 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 nonzero on a positive proportion of the integers. This improves on many previous works and is new already in the case of the Mobius 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.
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