Geometry of series terms of the Riemann Zeta Function

• Feb 21, 2023

• #1

Swamp Thing

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This is an Argand diagram showing the first 40,000 terms of the series form of the Riemann Zeta function, for the argument ##\sigma + i t = 1/2 + 62854.13 \thinspace i##

The blue lines are the first 100 (or so) terms, and the rest of the terms are in red. The plot shows a kind of approximate symmetry about the dotted green line, in the sense that the red part mirrors the blue part on average but is much more fine-grained. The last spiral in the red part is centred around the analytically continued value of zeta.

These aspects of the RZF seem rather interesting to me, but I could find only a limited amount of discussion about it --
1. https://arxiv.org/abs/1310.6396, G. Nickel, Geometry of the Riemann Zeta Function
2. https://arxiv.org/abs/1507.07631, G. Nickel. Symmetry in Partial Sums of the Riemann Zeta Function
3. https://ojs.stanford.edu/ojs/index.php/surj/issue/download/surj-2005/47, see pages 17-26.
4. http://laacademy.org/LAS2021/posters/P24_Conor_McGibboney.pdf

I am wondering why there isn't more interest in this. Is it because it is considered a dead end as far as research level mathematics?

Secondly, are there other sources that I have missed?

Thirdly, the symmetry property is not actually proved in the above references, but only explored phenomenologically. Is there a rigorous proof somewhere?

 

Last edited: Feb 21, 2023

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• Feb 22, 2023

• #2

martinbn

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Swamp Thing said:

...
I am wondering why there isn't more interest in this.
...

Shouldn't the question always be the opposite, why is there interest in this? What is the motivation?

 

• Feb 22, 2023

• #3

Swamp Thing

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Speaking only for myself -- curiosity.

It would make me happy to understand why the mirror symmetry emerges from this rather chaotic, almost Brownian behavior.

 

Last edited: Feb 22, 2023

• Mar 4, 2023

• #4

Swamp Thing

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I decided to ask ChatGPT, mentioning the G. Nickel papers that I linked to in my OP.

ChatGPT:

As far as I know, the exact reason for the observed mirror symmetry in the partial sums of the Riemann zeta function has not been fully explained or proven. Nickel himself acknowledges this in his papers, stating that the symmetry is "striking" but that "its origins are not well understood".

Some researchers have proposed various explanations, such as the role of modular symmetry, the connection to the Selberg trace formula, and the connection to random matrix theory. However, these explanations are still speculative and there is no consensus on the exact reason for the observed symmetry. The phenomenon remains an active area of research in number theory and related fields.

 

1. What is the Riemann Zeta Function?

The Riemann Zeta Function is a mathematical function that is defined for all complex numbers except 1. It is represented by the symbol ζ(s) and is named after the mathematician Bernhard Riemann who first studied its properties.

2. What is the significance of the Riemann Zeta Function?

The Riemann Zeta Function is important in number theory and has connections to other areas of mathematics such as algebra, calculus, and geometry. It is also used in physics and engineering to solve problems related to wave phenomena and quantum mechanics.

3. How is the Riemann Zeta Function related to the prime numbers?

The Riemann Zeta Function is closely related to the distribution of prime numbers. In fact, the Riemann Hypothesis, one of the most famous unsolved problems in mathematics, is a conjecture about the behavior of the Riemann Zeta Function and its connection to the prime numbers.

4. What is the Geometry of series terms of the Riemann Zeta Function?

The Geometry of series terms of the Riemann Zeta Function refers to the visual representation of the function as a series of points on a complex plane. This geometry can help us understand the behavior of the function and its properties.

5. How is the Riemann Zeta Function used in practical applications?

The Riemann Zeta Function has practical applications in fields such as cryptography, signal processing, and data compression. It is also used in the development of algorithms for solving complex mathematical problems and in the study of prime numbers and their distribution.