We consider the following family of random trigonometric polynomials form Sn(θ):=1 n ∑k=1nakcos(kθ)+bksin(kθ), where variables {ak,bk}k≥1 are i.i.d. and symmetric, defined on a common probability space (Ω,F,P). In seminal paper [SZ54], Salem Zygmund proved that for Rademacher coefficients, P−almost surely all t∈R limn→+∞ 1 2π∫02πeitSn(θ)dθ=e−t2∕2. To best our knowledge, natural question fluctua...