$\mathrm{\Omega }$-theorem for short trigonometric sum

We obtain in this paper new application of the classical E.C. Titchmarsh’ discrete method (1934) in the theory of the Riemann $\text{zf}$ - function. Namely, we shall prove the first localized $\mathrm{\Omega }$-theorem for short trigonometric sum. This paper is the English version of the work of reference \cite{4}.