A note on the Liouville function in short intervals

Abstract

In this note we give a short and self-contained proof that, for any $\delta >0$, ${\sum }_{x\le n\le x+x^{\delta }}\lambda (n)=o(x^{\delta })$ for almost all $x\in [X,2X]$. We also sketch a proof of a generalization of such a result to general real-valued multiplicative functions. Both results are special cases of results in our more involved and lengthy recent pre-print.

12 pages, expository note

Cited by in corpus (7)

• Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges
• Higher uniformity of bounded multiplicative functions in short intervals on average
• On the Möbius function in all short intervals
• Higher uniformity of arithmetic functions in short intervals I. All intervals
• Averages of the Möbius function on shifted primes
• The Liouville function in short intervals [after Matomaki and Radziwill]
• Averages of long Dirichlet polynomials