Alladi Ramakrishnan Hall

Sarnaks M ̈obius disjointness conjecture: ten year later!

EL Abdalaoui El Houcein

CNRS, France

Sarnaks M ̈obius disjointness conjecture assert that all dynamicalsystem with zero topological entropy satisfy the so-called M ̈obius randomnesslaw. This later law presume that the M ̈obius function changes sign randomly.It turns out that Sarnaks conjecture is related to Chowlas conjecture. Thislater conjecture predict that the Liouville function is normal.We recall that The Liouville function assigns the value +1 tonif thenumber of prime factors ofn, counted with multiplicities, is even and−1 ifnot. The M ̈obius function coincide with the Liouville function on the set ofsquare-free number and assigns the value zero otherwise. We recall that thenumbernis not square-free if there is a prime numberpsuch thatp2dividesnIn my talk, I will survey some recent works on Sarnaks conjecture andChowlas conjecture. I will also present my recent work on the equivalenceof these two conjectures, and my very recent joint work with M. Neruraker(Rutgers university, USA) in which we establish that the M ̈obius disjointnessconjecture holds for singular and Veech systems. As a consequence, we obtainsome improvement of Motohashi-Ramachandra result on the Mertens function in short interval.