Functional equation Riemann Zeta function

In summary, The Riemann functional equation has two forms, one of which is more symmetric and can be derived from the other and the duplication theorem of the Gamma function. There are two theorems related to this functional equation, with Corollary 2 being linked to Theorem 1 through the use of a different form of the equation. This can be seen by defining ##\xi(s)## using the formula in the corollary and using the functional formula from Theorem 1, resulting in ##\xi(s)=\xi(1-s)##.
  • #1
Lapidus
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There are two forms of Riemann functional equation. One is more symmetric and follows from the other and the duplication theorem of the Gamma function. At least, that's been claimed here:

https://terrytao.wordpress.com/2014...unction-and-the-functional-equation-optional/

Can someone help me linking Corollary 2 to Theorem 1?

I am just a amateurish layman who tries to piece things together from various sources from the net. So I got confused when I found out that people use two different forms of the functional equation.

Any help, hints or links are very much appreciated!
 
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  • #2
Another way to write the functional equation for ##\zeta## in Theorem 1 is

## \pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s)=\pi^{-\frac{1-s}{2}}\Gamma\left(\frac{1-s}{2}\right)\zeta(1-s) ##

so if you take as definition of ## \xi(s) ## the formula (6) in the corollary and using the functional formula above you can see that

## \xi(s)=\xi(1-s) ##
 
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1. What is the Riemann Zeta function?

The Riemann Zeta function is a mathematical function that was introduced by the mathematician Bernhard Riemann in the 19th century. It is defined as the infinite sum of 1/n^s, where n ranges from 1 to infinity and s is a complex number. This function is often denoted by the Greek letter ζ (zeta).

2. What is the functional equation of the Riemann Zeta function?

The functional equation of the Riemann Zeta function is a relation that connects the values of the function at s and 1-s. It is given by ζ(s) = 2^sπ^(s-1)sin(πs/2)Γ(1-s)ζ(1-s), where Γ is the gamma function. This equation allows us to extend the domain of the function to the entire complex plane, except for the point s=1.

3. What is the significance of the Riemann Zeta function in number theory?

The Riemann Zeta function is an important tool in number theory as it allows mathematicians to study the distribution of prime numbers. In particular, the function has zeros at negative even integers, which correspond to the prime numbers. This connection has been used to prove many important results in number theory, such as the prime number theorem.

4. Are there any real-world applications of the Riemann Zeta function?

The Riemann Zeta function has been used in various fields such as physics, engineering, and computer science. In physics, it has been used to model the behavior of particles in a quantum field. In engineering, it has been used to study the properties of electromagnetic fields. In computer science, it has been used in the development of algorithms for data compression and error correction.

5. What are some open problems related to the Riemann Zeta function?

One of the major open problems related to the Riemann Zeta function is the Riemann hypothesis, which states that all of the non-trivial zeros of the function lie on the line s=1/2. This has been a long-standing problem in number theory and its proof or disproof would have significant implications for the distribution of prime numbers. Other open problems include the distribution of zeros of the function and the behavior of the function at the critical line s=1.

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