Geometry of series terms of the Riemann Zeta Function

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  • #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##

1677033178046.png

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?
 
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  • #2
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?
 
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  • #3
Speaking only for myself -- curiosity.

It would make me happy to understand why the mirror symmetry emerges from this rather chaotic, almost Brownian behavior.
 
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  • #4
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.
 

FAQ: Geometry of series terms of the Riemann Zeta Function

What is the Riemann Zeta Function?

The Riemann Zeta Function, denoted as ζ(s), is a complex function defined for complex numbers s = σ + it, where σ and t are real numbers. It is initially defined for the real part σ > 1 by the infinite series ζ(s) = ∑(n=1 to ∞) n^(-s). The function has deep connections to number theory, particularly in the distribution of prime numbers, and can be analytically continued to other values of s, except for s = 1, where it has a simple pole.

What is meant by the geometry of series terms in the context of the Riemann Zeta Function?

The geometry of series terms refers to the study of the arrangement and behavior of the terms in the series representation of the Riemann Zeta Function in the complex plane. This includes examining how the terms n^(-s) behave as s varies, especially in relation to their magnitudes and phases, which can reveal insights into convergence, analytic properties, and the distribution of zeros of the function.

How do the series terms relate to the convergence of the Riemann Zeta Function?

The convergence of the series terms of the Riemann Zeta Function depends on the real part of s (σ). For σ > 1, the series converges absolutely, while for σ ≤ 1, the series diverges. The geometric interpretation of convergence can be visualized through the spatial distribution of the terms in the complex plane, where the distance from the origin and the angle can affect the convergence behavior.

What is the significance of the critical line in the context of the Riemann Zeta Function?

The critical line refers to the line in the complex plane where the real part of s is 1/2, specifically σ = 1/2. The Riemann Hypothesis posits that all non-trivial zeros of the Riemann Zeta Function lie on this critical line. The geometry of the series terms can provide insights into the behavior of the function along this line and help understand the distribution of its zeros.

How can visualizations of the Riemann Zeta Function enhance our understanding of its properties?

Visualizations of the Riemann Zeta Function, including plots of its magnitude and phase in the complex plane, can help elucidate the intricate relationships between its series terms and their geometric arrangement. Such visualizations can reveal patterns, symmetries, and anomalies in the function's behavior, aiding in the exploration of its analytic properties and the investigation of conjectures like the Riemann Hypothesis.

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