On the Möbius function in all short intervals

Abstract

We show that, for the Möbius function $\mu (n)$, we have $$\sum _{x<n\le x+x^{\theta }}\mu (n)=o(x^{\theta })$$ for any $\theta >0.55$. This improves on a result of Ramachandra from 1976, which is valid for $\theta >7/12$. Ramachandra's result corresponded to Huxley's $7/12$ exponent for the prime number theorem in short intervals. The main new idea leading to the improvement is using Ramaré's identity to extract a small prime factor from the $n$-sum. The proof method also allows us to improve on an estimate of Zhan for the exponential sum of the Möbius function as well as some results on multiplicative functions and almost primes in short intervals.

18 pages; referee comments incorporated

References in corpus (1)

• A note on the Liouville function in short intervals

Cited by in corpus (4)

• Higher uniformity of arithmetic functions in short intervals I. All intervals
• The Hardy--Littlewood--Chowla conjecture in the presence of a Siegel zero
• Möbius disjointness for a class of exponential functions
• Harper's beyond square-root conjecture