Jacob's ladders, $\zeta$-factorization and infinite set of metamorphosis of a multiform

TL;DR

This paper combines Jacob's ladders and classical formulas to establish a new $z$-factorization on the critical line, revealing control parameters for the metamorphosis of a multiform related to the Riemann-Siegel formula.

Contribution

It introduces a novel $z$-factorization formula on the critical line using Jacob's ladders and Hardy-Littlewood, and identifies control parameters for multiform metamorphosis linked to the Riemann-Siegel formula.

Findings

Proves a new $z$-factorization formula on the critical line.

Identifies control parameters for multiform metamorphosis.

Connects Jacob's ladders with classical number theory formulas.

Abstract

In this paper we use Jacob's ladders together with fundamental Hardy-Littlewood formula (1921) to prove the so-called $\zeta$-factorization formula on the critical line. Simultaneously, we obtain a set of control parameters of metamorphosis of a multiform connected with the Riemann-Siegel formula.