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

In summary, the conversation discusses the function Z(s) and its relation to the Riemann Hypothesis. It is stated that if the Riemann Hypothesis is true, then the roots of Z(s) would have a certain property. The conversation also mentions the Functional equation relating Z(1-s) and Z(s) and questions the existence of Z(s) for certain values of s. It is concluded that Z(s) exists for all values of s if Re(s) is greater than 1.
  • #1
zetafunction
391
0
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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  • #2
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
 
  • #3
True for Re s > 1, right? And therefore true for all s ...
 
  • #4
g_edgar said:
True for Re s > 1, right? And therefore true for all s ...

Yes, that's right. Thanks!

Petek
 

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

1. What is a Generalized ZETA function?

A Generalized ZETA function, also known as a Hurwitz ZETA function, is a mathematical function that generalizes the Riemann ZETA function. It is defined as ζ(s,a) = ∑(n=0 to ∞)(1/(n+a)^s), where s is a complex number and a is a positive real number.

2. What is the significance of the Generalized ZETA function?

The Generalized ZETA function has many applications in number theory, analysis, and physics. It is closely related to the Riemann ZETA function, which is important in the study of prime numbers. It also appears in the study of the distribution of primes, the Riemann hypothesis, and the theory of modular forms.

3. How is the Generalized ZETA function related to the Riemann ZETA function?

The Generalized ZETA function is a generalization of the Riemann ZETA function, meaning that the Riemann ZETA function is a special case of the Generalized ZETA function when a=1. The Riemann ZETA function is important in number theory, while the Generalized ZETA function has broader applications in other areas of mathematics and science.

4. Can the Generalized ZETA function be evaluated for all values of s and a?

No, the Generalized ZETA function is only defined for certain values of s and a. For example, it is defined for all complex values of s with Re(s)>1 and for all positive real values of a. However, it can be analytically continued to other values of s, allowing for its evaluation at those points as well.

5. Are there any famous open problems related to the Generalized ZETA function?

Yes, the Riemann hypothesis, one of the most famous unsolved problems in mathematics, is closely related to the Generalized ZETA function. It states that all non-trivial zeros of the Riemann ZETA function lie on the critical line Re(s)=1/2, which would also hold true for the Generalized ZETA function if it were true. Other open problems related to the Generalized ZETA function include the generalized Riemann hypothesis and the Lehmer conjecture.

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