Improve bounds of some arithmetical functions

الملخص EN

We show in this article the use of the norm function to get a new lower bound of Riemann-Zeta function where .

This subject has been studied deeply by Hilberdink [HIL, 12]).

Getting a bound for the Riemann-Zeta function in the critical strip is more challenging for many reasons related to the behavior of the Riemann-Zeta function in that strip.

In the other words, the aim of this article is to prove that has a strict lower bound when the real part is very closed to the line 1.

We state this in the main theorem of this paper.

Key words: Analytic Number Theory (especially, The Riemann-Zeta function), Banach space and the norm function.-