http://uk.answers.yahoo.com/question/index?qid=20130120221522AAKHPCD
Is Riemann Hypothesis based on Golden key?
This question references [stapler] comment to ( http://uk.answers.yahoo.com/question/index?qid=20130115145523AAtC9Ol )
"That's exactly what I mean. The Riemann Hypothesis is not based on the Euler product."
I disagree and let me explain why:
Golden key is not Euler product.
Golden key is the relationship (equation) of Euler product to Riemann zeta function
(Euler product = Riemann zeta function)
"That's exactly what I mean. The Riemann Hypothesis is not based on the Euler product."
I disagree and let me explain why:
Golden key is not Euler product.
Golden key is the relationship (equation) of Euler product to Riemann zeta function
(Euler product = Riemann zeta function)
http://en.wikipedia.org/wiki/Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function
- Euler product formula is constructed as a multiplication using prime numbers(p).
- Riemann zeta function is constructed as a sum using a series of integers(n)/complex variables(s).
Riemann Hypothesis is directly based on Riemann zeta function with limitations (non-trivial zeros, critical line)
Riemann Hypothesis is about an expected pattern of prime numbers using Riemann zeta function.
Since, Golden key defines the relationship of Riemann zeta function to prime numbers, Riemann Hypothesis (assumed pattern for prime numbers) is based on Golden key, and in-turn, based on Euler product as well.
Even, there cannot be a suggestion as Riemann Hypothesis, if Golden key (relationship to prime numbers) did not exist at that time.
Though Euler product is perfectly true, it's relationship(equation) to Riemann zeta function called 'The Golden Key' is an Error Proof. I am sure about that.
Therefore, Riemann Hypothesis is based on a false Golden key and prime number pattern cannot be derived from Riemann zeta function in any condition.
The stage is open for you to comment and criticize.
My challenges to you:
1. prove Riemann Hypothesis is not based on Golden key
2. prove Golden key is true
3. prove Riemann Hypothesis is on firm grounds and ClayMath Millennium problem is valid
You may reference to your work and also may contact me personally if you're serious on this matter.
Riemann Hypothesis is about an expected pattern of prime numbers using Riemann zeta function.
Since, Golden key defines the relationship of Riemann zeta function to prime numbers, Riemann Hypothesis (assumed pattern for prime numbers) is based on Golden key, and in-turn, based on Euler product as well.
Even, there cannot be a suggestion as Riemann Hypothesis, if Golden key (relationship to prime numbers) did not exist at that time.
Though Euler product is perfectly true, it's relationship(equation) to Riemann zeta function called 'The Golden Key' is an Error Proof. I am sure about that.
Therefore, Riemann Hypothesis is based on a false Golden key and prime number pattern cannot be derived from Riemann zeta function in any condition.
The stage is open for you to comment and criticize.
My challenges to you:
1. prove Riemann Hypothesis is not based on Golden key
2. prove Golden key is true
3. prove Riemann Hypothesis is on firm grounds and ClayMath Millennium problem is valid
You may reference to your work and also may contact me personally if you're serious on this matter.
Answer:
First, your "definition' (1) of the zeta function is incorrect; the equality (1) only holds for Re(s)>1. Given that the RH concerns Re(s) = 1/2, you have to take into account the analytic continuation of (1).
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