Jacob's ladders and the asymptotic formula for short and microscopic parts of the Hardy-Littlewood integral of the function $|\zeta(1/2+it)|^4$

TL;DR

This paper introduces new asymptotic formulas for short and microscopic segments of the Hardy-Littlewood integral of the Riemann zeta function, using geometric properties of Jacob's ladders, which are not accessible by traditional methods.

Contribution

It presents novel asymptotic formulas derived from geometric properties of Jacob's ladders, expanding the analytical tools for studying the zeta function.

Findings

New asymptotic formulas for short parts of the Hardy-Littlewood integral

Formulas applicable to microscopic segments of the integral

Methods differ from classical approaches of Balasubramanian, Heath-Brown, and Ivic

Abstract

The elementary geometric properties of Jacob's ladders of the second order lead to a class of new asymptotic formulae for short and microscopic parts of the Hardy-Littlewood integral of $|\zeta(1/2+it)|^4$. These formulae cannot be obtained by methods of Balasubramanian, Heath-Brown and Ivic.