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Published April 19, 2016 | Published + Submitted
Journal Article Open

Multiplicative functions in short intervals


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 Möbius function we show that there are cancellations in the sum of μ(n) in almost all intervals of the form [x,x+ψ(x)] with ψ(x)→∞ 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 xε-smooth numbers in intervals of the form [x,x+c(ε)√x], recovering unconditionally a conditional (on the Riemann Hypothesis) result of Soundararajan. Third, we show that the mean-value of λ(n)λ(n+1), with λ(n) Liouville's function, is nontrivially bounded in absolute value by 1–δ for some δ>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 Möbius 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.

Additional Information

© 2016 Department of Mathematics, Princeton University. Received: 6 March 2015; Revised: 18 September 2015; Accepted: 23 October 2015; Published online: 19 April 2016. The authors would like to thank Andrew Granville for many useful discussions on the topic. They would also like to thank the anonymous referee and Joni Teräväinen for careful reading of the manuscript. The first author was supported by the Academy of Finland grants no. 137883 and 138522.

Attached Files

Published - annals-v183-n3-p06-p.pdf

Submitted - 1501.04585.pdf


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August 20, 2023
October 18, 2023