Power series with random and pseudo-random coefficients
Location
Study of the influence of the arguments of the coefficients of power series on their zero distribution was initiated long time ago by Levy, Littlewood, and Offord and still remains a terra incognita within the complex analysis.
Our findings are: (a) the asymptotics of the counting function of zeroes of Rademacher power series (the main ingredient is the logarithmic integrability of Fourier series with random signs), and (b) the intimate relation between the zero distribution and the pair correlations (the Wiener spectrum) of the coefficients, which allows us to treat examples of a very different nature (stationary ergodic sequences, Besicovitch almost-periodic sequences, and sequences of arithmetic origin).
The talk is based on joint works with Jacques Benatar, Alexander Borichev, Fedor Nazarov, and Alon Nishry.