Jake Chinis

University of Bristol

On the Liouville function in short intervals

Linfoot Number Theory Seminar

27th September 2023, 11:00 am – 12:00 pm
Fry Building, 2.04

From the seminal work of Matom{\"a}ki and Radziwi{\l}{\l} on multiplicative functions in short intervals, we know that the "local" averages of all one-bounded multiplicative functions are equal to the corresponding "global" averages almost always. In the case of the Liouville function, $\lambda$, their work can be rephrased as follows: the partial sums of $\lambda$ exhibit some cancellation in almost all intervals of the form $[x,x+h]$, for any $h=h(x)$ going to infinity arbitrarily slowly. Assuming Riemann's Hypothesis (RH), we further know that the partial sums of the Liouville function exhibit square-root cancellation (globally). This begs the following question: assuming RH, do the partial sums of $lambda$ exhibit square-root cancellation in almost all short intervals? In this talk, we answer the question in the affirmative, provided that the length of the interval is neither too large, nor too small. The proof uses a simple variant of the methods developed by Matom{\"a}ki and Radziwi{\l}{\l}, as well as some standard results on integers free of large prime factors.

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