  We improve the current bounds for an inequality of Erd\H{o}s and Tur\'an from
1950 related to the discrepancy of angular equidistribution of the zeros of a
given polynomial. Building upon a recent work of Soundararajan, we establish a
novel connection between this inequality and an extremal problem in Fourier
analysis involving the maxima of Hilbert transforms, for which we provide a
complete solution. Prior to Soundararajan (2019), refinements of the
discrepancy inequality of Erd\H{o}s and Tur\'an had been obtained by Ganelius
(1954) and Mignotte (1992).
