Analytically Continue RZF Using Gamma Function: Step By Step Guide

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In summary, the conversation discusses the theory of the Riemann zeta function and its analytical continuation using the gamma function. It also mentions the Jacobi theta function and Riemann's use of the functional equation to equate two expressions. The main focus is on resolving an error in the book that caused confusion in understanding the last equation in Riemann's manuscript.
  • #1
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theory of Riemann zeta function question

analytically continuing the Riemann zeta function (RZF) using the gamma function leads to this identify:

[tex]n^{-s} \pi^{-s \over 2} \Gamma ({s \over 2}) = \int_0^{\infty} e^{-n^2 \pi x} x^{{s \over 2}-1} dx[/tex] ________(1)

and from that we can build a similar expression incorporating the RZF:

[tex]\pi^{-s \over 2} \Gamma ({s \over 2}) \zeta (s) = \int_0^{\infty} \psi (x) x^{{s \over 2}-1} dx[/tex] ________(2)

where

[tex]\psi (x) = \sum_{n=1}^{\infty} e^{-n^2 \pi x}[/tex]

is the Jacobi theta function.

Then Riemann proceeds to use the functional equation for the theta function:

[tex]2\psi (x) +1 = x^{-1 \over 2} (2 \psi ({1 \over x})+1)[/tex]

to equate (2) with:

[tex]\pi^{-s \over 2} \Gamma ({s \over 2}) \zeta (s) = \int_1^{\infty} \psi (x) x^{{s \over 2}-1} dx + \int_1^{\infty} \psi ({1 \over x}) x^{-1 \over 2} x^{{s \over 2}-1} dx+{1 \over 2} \int_0^{1} (x^{-1 \over 2} x^{{s \over 2}-1} - x^{{s \over 2}-1}) dx[/tex]

This is the step I am stuck on, I am trying to figure out what he did to get that last equation. Any help would be greatly appreciated.
 
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  • #2
I found an error, and the reason why I've been stuck. There is a typo in my book, maybe I should contact the publisher. I had to check Riemann's original handwritten manuscript in german to see that the second integral is from 0 to 1.

the second integral is not

[tex]\int_1^{\infty} \psi ({1 \over x}) x^{-1 \over 2} x^{{s \over 2}-1} dx[/tex]

but

[tex]\int_0^1 \psi ({1 \over x}) x^{-1 \over 2} x^{{s \over 2}-1} dx[/tex]
 
  • #3
problem resolved. I verified and indeed

[tex]\pi^{-s \over 2} \Gamma ({s \over 2}) \zeta (s) = \int_1^{\infty} \psi (x) x^{{s \over 2}-1} dx + \int_0^1 \psi ({1 \over x}) x^{-1 \over 2} x^{{s \over 2}-1} dx+{1 \over 2} \int_0^{1} (x^{-1 \over 2} x^{{s \over 2}-1} - x^{{s \over 2}-1} ) dx[/tex]

=>

[tex]\zeta (s) = {\pi^{s \over 2} \over \Gamma (\frac{s}{2})} \left( \int_1^{\infty} \psi (x) x^{{s \over 2}-1} dx + \int_0^1 \psi ({1 \over x}) x^{-1 \over 2} x^{{s \over 2}-1} dx+{1 \over 2} \int_0^{1} (x^{-1 \over 2} x^{{s \over 2}-1} - x^{{s \over 2}-1} ) dx \right)[/tex]

=>

[tex]\zeta (s) = {\pi^{s \over 2} \over \Gamma (\frac{s}{2})} \left( {1 \over s(s-1)} + \int_1^\infty \psi(x) \left( x^{{s \over 2}-1} + x^{-{s+1 \over 2}} \right) dx \right)[/tex]
 
  • #4
The error I am talking about can be found in the middle of page 3 not counting the cover page.
http://www.claymath.org/millennium/Riemann_Hypothesis/1859_manuscript/EZeta.pdf [Broken]


Riemann's manuscript, bottom of page 2, where you can see the real version.
http://www.claymath.org/millennium/Riemann_Hypothesis/1859_manuscript/riemann1859.pdf [Broken]
 
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  • #5


I can say that this is a very interesting and complex topic in mathematics. The Riemann zeta function is a fundamental object in number theory and its analytical continuation has important implications in various areas of mathematics and physics. The use of the gamma function in this context is a powerful tool that allows us to extend the domain of the RZF and study its behavior in a wider range of values.

In this step-by-step guide, you have shown the process of analytically continuing the RZF using the gamma function and deriving the identity (1). This identity is then used to build a similar expression (2) incorporating the RZF. However, to equate (2) with the functional equation for the theta function, Riemann makes use of a clever substitution and some algebraic manipulation to arrive at the last equation.

This type of mathematical reasoning and manipulation is often necessary in order to make connections and derive new identities. It is a crucial part of the scientific process and shows the creativity and ingenuity of mathematicians in solving complex problems.

Overall, this guide provides a clear and detailed understanding of the steps involved in analytically continuing the Riemann zeta function using the gamma function. It is a valuable resource for anyone interested in this topic and showcases the beauty and complexity of mathematical concepts.
 

1. What is Analytically Continue RZF using Gamma Function?

Analytically Continue RZF (Riemann Zeta Function) using Gamma Function is a mathematical method used to extend the Riemann Zeta function to the entire complex plane. This method uses the properties of the Gamma function to analytically continue the Riemann Zeta function, which is defined only for real numbers greater than 1, to the entire complex plane.

2. Why is Analytically Continue RZF using Gamma Function important?

The Riemann Zeta function is an important tool in number theory and has applications in various fields of mathematics. By extending the Riemann Zeta function to the entire complex plane, we can better understand its properties and use it in a wider range of mathematical problems.

3. What are the steps involved in Analytically Continue RZF using Gamma Function?

The steps involved in Analytically Continue RZF using Gamma Function include defining the Riemann Zeta function, using the Gamma function to analytically continue the Riemann Zeta function, and then using this extended function to solve mathematical problems. This process can be complex and often requires advanced mathematical knowledge.

4. What are the applications of Analytically Continue RZF using Gamma Function?

Analytically Continue RZF using Gamma Function has various applications in mathematics, including in number theory, complex analysis, and physics. It is also used in solving problems related to the distribution of prime numbers and the Riemann Hypothesis, which is one of the most famous unsolved problems in mathematics.

5. Are there any limitations to Analytically Continue RZF using Gamma Function?

While Analytically Continue RZF using Gamma Function has many applications, it also has some limitations. This method can only be applied to certain types of functions, and the process can be complex and time-consuming. Additionally, the accuracy of the results may depend on the initial assumptions made during the process.

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