What Happens to the Roots of Z(s) if the Riemann Hypothesis Holds True?

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Discussion Overview

The discussion revolves around the implications of the Riemann Hypothesis on the roots of the function Z(s), defined as the infinite product of the Riemann Zeta function, and the conditions under which Z(s) exists. Participants explore the relationship between Z(s) and the Riemann functional equation.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions what happens to the roots of Z(s) if the Riemann Hypothesis holds true, specifically regarding the functional equation relating Z(1-s) and Z(s).
  • Another participant raises the issue of convergence for the infinite product defining Z(s), suggesting that individual terms must converge to 1 for Z(s) to exist.
  • There is a claim that the condition for convergence holds true for Re s > 1, although the implications for all s are not fully explored.
  • A later reply confirms the condition for Re s > 1, but does not clarify its applicability to other values of s.

Areas of Agreement / Disagreement

Participants generally agree on the condition for convergence of Z(s) for Re s > 1, but there is uncertainty regarding the implications for other values of s and the overall relationship to the Riemann Hypothesis.

Contextual Notes

The discussion does not resolve the specific conditions under which Z(s) exists beyond Re s > 1, nor does it clarify the functional equation relating Z(1-s) and Z(s).

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given the function [tex]Z(s)= \prod _{k=0}^{\infty}\zeta (s+k)[/tex] with [tex]\zeta (s)[/tex] being the Riemann Zeta function

the idea is if ALL the roots have real part (i mean Riemann Hypothesis) is correct, then what would happen with the roots of Z(s) ??

what would be the Functional equation relating Z(1-s) and Z(s) ¿¿ from the definition of Riemann functional equation
 
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For which values of s does Z(s) exist? A necessary condition for an infinite product to converge is that the individual terms converge to 1. Thus, it must be true that

[tex]\zeta(s + k) \rightarrow 1[/tex] as [tex]k \rightarrow \infty[/tex]

It's not clear to me for which values of s this holds.

Petek
 
True for Re s > 1, right? And therefore true for all s ...
 
g_edgar said:
True for Re s > 1, right? And therefore true for all s ...

Yes, that's right. Thanks!

Petek
 

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