Uniformity of the M\\\"obius function in $\\mathbf{F}_q$


Thai Hoang Le ( University of Mississippi, USA )
2019-01-07  10:30 - 11:30
Room 103, Mathematics Research Center Building (ori. New Math. Bldg.)

The M\\\"obius randomness principle states that the M\\\"obius function $\\mu$ does not correlate with simple or low complexity sequences $F(n)$, that is, we have non-trivial bounds for sums $\\sum_{n=1}^N \\mu(n) F(n)$. By analogy between the integers and the ring $\\mathbf{F}_q[t]$ of polynomials over a finite field $\\mathbf{F}_q$, we study this principle in the latter setting and expect that for $f$ in $\\mathbf{F}_q[t]$, $\\mu(f)$ does not correlate with low degree polynomials evaluated at the coefficients of $f$. In this talk, I will talk about our results in the linear and quadratic cases. Our main tool is a new result in additive combinatorics, namely a bilinear version of the Bogolyubov theorem. This is joint work with Pierre-Yves Bienvenu.