Riemann Zeta Function Zeros

In summary, the Wolfram Alpha page on the Riemann Zeta Function Zeros includes three graphs with a hole through each one. The third axis represents the plotted value and includes the real and imaginary parts, as well as the absolute value of the Zeta function. It is unclear if the hole in the graph is a nontrivial zero or a result of the function.
  • #1
SpaceDreamer
45
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I was looking at the Wolfram Alpha page on the Riemann Zeta Function Zeros which can be found here, http://mathworld.wolfram.com/RiemannZetaFunctionZeros.html

At the top of the pag there are three graphs each with what looks to be a hole through the graph. Now I know the graph is an Argand Diagram showing the function in bot the real and imaginary axis but what is the third axis? Another question is for the graph on the right the
l ζ(z) l which is the absolute value of the Zeta function correct? If it is not the Zeta function then what is the equation?

Is the hole in the graph a nontrivial zero or just a result of the function?

Thank You.
 
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  • #2
The third axis is the plotted value - the real part, the imaginary part, and the absolute value of the Zeta function.

Is the hole in the graph a nontrivial zero or just a result of the function?
The values are too large to get displayed, that looks like a pole.
 

1. What is the Riemann Zeta Function?

The Riemann Zeta Function, denoted as ζ(s), is a mathematical function used to study the distribution of prime numbers. It was first introduced by Bernhard Riemann in 1859 and is defined as the sum of the reciprocals of all positive integers raised to the power of s.

2. What are the key properties of the Riemann Zeta Function?

Some of the key properties of the Riemann Zeta Function include its analytic continuation and functional equation. It is also closely related to the distribution of prime numbers and has connections to other areas of mathematics such as number theory and physics.

3. What are Riemann Zeta Function zeros?

Riemann Zeta Function zeros are the complex numbers s for which ζ(s) = 0. These zeros are important in the study of the distribution of prime numbers and have important connections to the Riemann Hypothesis, which states that all non-trivial zeros of the Riemann Zeta Function lie on the critical line with real part 1/2.

4. How many Riemann Zeta Function zeros are there?

The Riemann Zeta Function has an infinite number of zeros, as it has infinitely many non-trivial zeros on the critical line. However, the exact number of zeros is unknown and is a topic of ongoing research.

5. What are the practical applications of the Riemann Zeta Function?

The Riemann Zeta Function has many practical applications in number theory, physics, and engineering. It has been used to study the distribution of prime numbers, construct efficient error-correcting codes, and even to solve problems related to quantum mechanics. It also has connections to other areas of mathematics such as fractals and chaos theory.

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