Number Theory Seminar: On the Báez-Duarte criterion for the Riemann hypothesis

Define $e_{n}(t)=\{t/n\}$. Let $d_N$ denote the distance in $L^2(0,\infty ; t^{-2}dt)$ between the indicator function of $[1,\infty[$ and the vector space generated by $e_1, \dots, e_N$. A theorem of B\'aez-Duarte (2003) states that the Riemann hypothesis (RH) holds if and only if $d_N \rightarrow 0$ when $N \rightarrow \infty$. Assuming RH, we prove the estimate $$d_N^2 \leq (\log \log N)^{5/2+o(1)}(\log N)^{-1/2}.$$ I shall put this result in its historical context, from Nyman's criterion (1950) and its beautiful proof to a sketch of our proof. I shall focus on the main ingredient we used, a method of Maier and Montgomery, recently sharpened by Soundararajan, to get some upper bound for partial sums of the M\"obius function. (joint work with Michel Balazard)