Entire functions of exponential type represented by pseudo-random and random Taylor series
We study the influence the angular distribution of zeroes of the Taylor
series with pseudo-random and random coefficients, and show that the
distribution of zeroes is governed by certain autocorrelations of the
coefficients. Using this guiding principle, we consider several examples
of random and pseudo-random sequences $\xi$ and, in particular, answer
some questions posed by Chen and Littlewood in 1967.
As a by-product we show that if $\xi$ is a stationary random
integer-valued sequence, then either it is periodic, or its spectral
measure has no gaps in its support. The same conclusion is true if $\xi$
is a complex-valued stationary ergodic sequence that takes values from a
uniformly discrete set (joint work with Alexander Borichev and Alon Nishry).