The Epstein Zeta Function: Definition, Zeros, and Functional Equation Explained

In summary, the Epstein zeta function is a mathematical function that has complex zeros and satisfies a functional equation given by (1). It is closely related to the Riemann zeta function and the Riemann hypothesis may also hold for the Epstein function. Further research is needed to fully understand its properties and implications.
  • #1
eljose
492
0
Hello, i would like to know the definiton of epstein zeta function and its zeros,i would like to know if s is a zero then s* is also a zero and what would be the functional equation that the Epstein function satisfy...thanks.
 
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  • #3
Thanks..for the links but i have not been able to see a link that gives you the functional equation for Epstein series...

EDIT: i have finally found the functional equation for Epstein Zeta function [tex]Zeta_{Q}(s)[/tex] in the form:

[tex]\pi^{-s}\Gamma(s)Z_{Q^{-1}}(s)=|Q|^{1/2}\pi^{s-n/2}\Gamma(n/2-s)Z_{Q}(n/2-s) [/tex] (1)

with |Q|=Det(Q), where for n=2 Q is the matrix of the quadratic form Q(x,y)=ax^2+by^2+cxy, with n=2 and Q=Q^{-1} then |Q|=1 for the cases n=2 and Q=Q^{-1}=R the functional equation (1) is exactly equal to Riemann functional equation for the function [tex]\zeta(s)[/tex] my question is if RH would hold also for this case of the Epstein function as for the case of the Riemann zeta function.
 
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1. What is the Epstein Zeta function?

The Epstein Zeta function is a mathematical function that was first introduced by Paul Epstein in 1903. It is an extension of the Riemann zeta function and is defined as a series of infinite terms involving the Hurwitz zeta function.

2. How are the zeros of the Epstein Zeta function related to the Riemann Hypothesis?

The Riemann Hypothesis states that all non-trivial zeros of the Riemann zeta function lie on the critical line with a real part of 1/2. Similarly, the Epstein Zeta function also has its non-trivial zeros on the critical line, making it closely related to the Riemann Hypothesis.

3. What is the functional equation of the Epstein Zeta function?

The functional equation of the Epstein Zeta function relates the values of the function at s and 1-s. It can be expressed as Z(s) = Z(1-s) * E(s), where E(s) is a term that depends on the parameters of the function.

4. What are some applications of the Epstein Zeta function?

The Epstein Zeta function has applications in number theory, algebraic geometry, and physics. It can be used to study the distribution of prime numbers, the arithmetic of elliptic curves, and the spectral theory of quantum systems, among others.

5. Does the Epstein Zeta function have any real-world implications?

While the Epstein Zeta function is an important tool in mathematics, it does not have any direct real-world implications. However, its study has led to a better understanding of the Riemann Hypothesis and other related problems, which have practical applications in fields such as cryptography and coding theory.

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