$\Omega$-theorem for short trigonometric sum

TL;DR

This paper applies Titchmarsh's discrete method to establish a localized -theorem for short trigonometric sums related to the Riemann zeta function, advancing understanding of its oscillatory behavior.

Contribution

It introduces the first localized -theorem for short trigonometric sums using classical discrete methods in the context of the Riemann zeta function.

Findings

Proves a new localized -theorem for short trigonometric sums.

Demonstrates the effectiveness of Titchmarsh's discrete method in this setting.

Provides insights into the oscillatory nature of the Riemann zeta function.

Abstract

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