Taussky-Todd Memorial Lecture in Mathematics

The Riemann Hypothesis on zeros of the ζ-function is known to be equivalent to a completeness of the (non-periodic) dilation system (ϕ(nx))_{n≥1} in the standard Lebesgue space L^2(0,1) for a rather particular generating function ϕ (L. Báez-Duarte (2003), A. Beurling, B. Nyman (1950)).

A general dilation completeness problem (for systems (ϕ(nx))_{n≥1} in L^2(0,1) was raised by Aurel Winter and Arne Beurling in 1940ies and remains a challenging open problem. Using an approach with the so-called "Bohr lift techniques" we show that for 2-periodic functions ϕ the problem is equivalent to the question on cyclic vectors of the Hardy space H^2 D^∞_2 on the infinite dimensional Hilbert multidisc D^∞_2. The results obtained on this way include practically all previously known results in the dialation completeness problem.