We prove that the logarithm of Fourier series with random signs is integrable to any positive power. Using this result we answer a long-standing question by J.-P. Kahane, concerning the range of random power series in the unit disk. In addition, we use this result to prove the angular equidistribution of the zeros of entire functions with random signs.

This is a joint work with F. Nazarov and M. Sodin.