Seminario del 2022

2022
11 aprile
Francesco Cellarosi (Queen's University, Canada)
Seminario interdisciplinare
The Möbius function \mu plays a central role in Number Theory. If n is not square-free (i.e. it is divisible by the square of some prime), then \mu(n)=0 otherwise \mu(n) equals +1 or -1 depending on the parity of the number of prime divisors of n. The average behaviour of this function can be understood by considering its partial sums. The problem of estimating the growth of such sums can be can easy (equivalent to the Prime Number Theorem) or very hard (equivalent to the Riemann Hypothesis), depending on the precision we require. Understanding the `randomness’ of the Möbius function can done by studying its autocorrelations (conjectured to be all zero by Chowla in 1965) or its correlations with other sequences. In 2010 Sarnak conjectured that the Möbius function should not correlate with any sequence of low complexity, i.e. sequences generated by dynamical systems with zero topological entropy. We will discuss what is known about Chowla’s and Sarnak’s conjectures and some of their weaker forms. We can ask to what extent the Möbius function behaves like a sequence of random variables with values in {0,+1,-1}, but we cannot hope for independence. In fact, when we study the simpler sequence \mu^2 (which is the indicator of the set of square-free integers) we see that it highly self-correlated. It can be shown, in fact, that \mu^2 is a typical realization of a stochastic process with as little randomness as possible. The approach we take in the study of such problem is dynamical, which has proven very fruitful. Time permitting, we will also survey some very recent results on the statistics of square-free integers in short intervals, where randomness re-appears.

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