On the M\"obius function in all short intervals

TL;DR

This paper proves that the sum of the Möbius function over short intervals of length x^θ is negligible for θ>0.55, improving previous bounds and employing Ramaré's identity to refine estimates in number theory.

Contribution

It introduces a new approach using Ramaré's identity to improve bounds on Möbius sums in short intervals, surpassing the classical 7/12 exponent.

Findings

Sum of μ(n) over short intervals is o(x^θ) for θ>0.55

Improved bounds on exponential sums of the Möbius function

Enhanced estimates on multiplicative functions and almost primes in short intervals

Abstract

We show that, for the M\"obius function $\mu(n)$, we have $$ \sum_{x < n\leq x+x^{\theta}}\mu(n)=o(x^{\theta}) $$ 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 Ramar\'e'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 M\"obius function as well as some results on multiplicative functions and almost primes in short intervals.