Appears in collection : Bourbaki - Juin 2016

The Liouville function λ(n) is a completely multiplicative function, taking the value 1 if n has an even number of prime factors (counted with multiplicity) and −1 if n has an odd number of prime factors. This function is expected to behave like a “random” collection of signs, plus or minus one both being equally likely. For example, a famous conjecture of Chowla asserts that the values of λ(n) and λ(n + 1) (and more generally translates of any k fixed distinct integers) are uncorrelated. Another well known belief was that almost all intervals with length tending to infinity should have roughly an equal number of plus and minus values of the Liouville function. Recently, Matomäki and Radziwiłł established that this last belief is indeed true, and more generally established a variant of such a result for a general class of multiplicative functions. Further joint work with Tao led to the proof of average versions of the Chowla conjecture, and to proving the existence of new sign patterns in the Liouville function. Finally, the recent work of Tao establishes a logarithmic version of the Chowla conjecture, and building on this settled the Erdös discrepancy conjecture. I will discuss some of the ideas behind these results in the Seminar.

[After Matomäki and Radziwiłł]