Riemann zeta function generalization

In summary, the Riemann zeta function has a generalization known as the Hurwitz zeta function, which includes a parameter h and can be extended to include negative values of s. This generalization is part of a larger world of generalized zeta functions, many of which have been proven to satisfy the generalized Riemann hypothesis.
  • #1
lokofer
106
0
"Riemann zeta function"...generalization..

Hello my question is if we define the "generalized" Riemann zeta function:

[tex] \zeta(x,s,h)= \sum_{n=0}^{\infty}(x+nh)^{-s} [/tex]

which is equal to the usual "Riemann zeta function" if we set h=1, x=0 ,then my question is if we can extend the definition to include negative values of "s" (using a functional equation or something similar)..:-p :-p
 
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  • #2
[tex] \zeta(x,s,h)= \sum_{n=0}^{\infty}(x+nh)^{-s}=h^{-s} \sum_{n=0}^{\infty}(x/h+n)^{-s}[/tex]

It's just a Hurwitz zeta function.
 
  • #3
There is a whole well documented world of things like this out there, Jose. L functions, generalized zeta functions, indeed the generalized Riemann hypothesis is known to be true for many many of the generalized zeta functions.
 

Related to Riemann zeta function generalization

1. What is the Riemann zeta function and why is it important?

The Riemann zeta function, denoted by ζ(s), is a mathematical function that was first introduced by Bernhard Riemann in the 19th century. It is defined as the infinite sum of the reciprocals of all positive integers raised to the power of s. This function plays a crucial role in number theory and has connections to other areas of mathematics, such as complex analysis and algebraic geometry.

2. Can the Riemann zeta function be generalized to other complex numbers?

Yes, the Riemann zeta function can be generalized to complex numbers outside the critical strip (the region where the real part of s is between 0 and 1). This is known as the completed zeta function and is denoted by ζ(s). It has applications in studying the distribution of prime numbers and has connections to the Riemann hypothesis.

3. What is the significance of the Riemann hypothesis in relation to the zeta function?

The Riemann hypothesis is a conjecture that states all non-trivial zeros of the Riemann zeta function lie on the critical line with a real part of 1/2. This hypothesis has been one of the most famous unsolved problems in mathematics and has important implications for the distribution of prime numbers. Many other zeta functions have their own versions of the Riemann hypothesis, making it a widely studied topic.

4. Are there other generalizations of the Riemann zeta function?

Yes, there are many other generalizations of the Riemann zeta function, some of which include the Dirichlet zeta function, the Hurwitz zeta function, and the Epstein zeta function. Each of these functions has its own unique properties and applications in mathematics, such as in number theory and physics.

5. How does the Riemann zeta function relate to the prime number theorem?

The prime number theorem is a fundamental result in number theory that provides an estimate for the number of prime numbers less than a given number. The Riemann zeta function plays a key role in proving this theorem, as the behavior of the zeta function at certain points is closely related to the distribution of prime numbers. The prime number theorem and the Riemann hypothesis are also intimately connected through the use of the zeta function.

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