A note on the Liouville function in short intervals

2015/02/09 by Kaisa Matomäki, Maksym Radziwiłł, Matomäki, Kaisa +1 · 1 citation
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1502.02374

openalex publication_date 2015/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note we give a short and self-contained proof that, for any δ> 0, ∑_{x ≤ n ≤ x+x^δ} λ(n) = o(x^δ) for almost all x ∈ [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.

Citations

• Multiplicative functions in short intervals

Cited by

• The logarithmically averaged Chowla and Elliott conjectures for two-point correlations

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