The Beurling--Wintner problem and analytic number theory

This talk concerns a long-standing problem on completeness of function systems generated by odd periodic extensions  of  functions in L^2(0,1). This problem, raised by Beurling and Wintner in the 1940s,  is closely related to the Riemann Hypothesis. We completely solve the rational version of step functions (that is,  those functions with  rational jump discontinuities) by approaches from analytic number theory, and present several deep applications including a complete  solution to the rational version of Kolzov completeness problem. This is a joint work with Dr. Hui Dan.