Zeta Function Puns

I am struggling to understand how to make sense of how the plotting works re. Riemann's Zeta Function, and specifically using prime numbers.

Basically, every time I hear the phrase "zeta zeros", I get confused when trying to make sense of it all. There are the "trivial zeros" and the "non trivial zeros" for the "zeta zeros" I think.

As I understand it, there are these two interesting ways to plot the numbers.

1. Plotting points on the critical line with real value 0,5, inside the critical strip between 0 and 1. Presumably the plotted points are converging sums, taking the zeta function to infinity so to speak.

2. Plotting points, and interpolating them, creating a curved line that winds around origo, but crosses origo perfectly on every turn. This, as I understand it, the curve repeatedly connects to zero, or origo.

I wanted to figure out how to work with prime numbers in all of this, but, perhaps somebody please could comment on my notions for the two ways to plot the points and how it all makes sense. I know that there is something called 'analytic continuation', but how that related to the two ways to plot the points I wouldn't know.

I am curious to learn how to plot non trivial zeros on the critical line, but this can wait a little if there are other things I have to understand first.

What I mean is:

does a number x exist which satisfies:

ζ(x)=x (where ζ(x) is the zeta function)

and if so, is it trancendetal? or at least, what are some of the properties that it has?

It just seems really cool to me, and wolframalpha's answers don't really satisfy me because there isn't any infromation about them except their value.

Thank you☺

The sums [sum_{s=1}^\infty 1/n^s] appear frequently in basic calculus talking about convergence; they seem to be the prototypical example of series whose convergence can be (most) easily studied using the corresponding integral.

However, it looks to me right now to be a remarkable leap to extend this to the complex plane. Can someone illuminate Riemann's thought process, or was this move simply a stroke of genius that us mere mortals could not hope to ever reproduce?

Note: I haven't taken complex analysis yet, so correct me if I'm wrong on anything

Visually the circular arches and their symmetry around the line Re(s)=γ in the Reimann Zeta Transformation are probably the most prominent feature of the transformation. /u/direwolf202 provided a proof here as to why the graph appears to be symmetrical around γ. I believe the limit he references (lim s -> 1 (ζ(s) - 1/(s-1)) = γ) also illustrates another fact. Around the pole s=1, ζ(s) behaves very similarly to the complex function f(s)=1/(s-1) + γ. This behavior can also be explained by the Laurent series expansion of ζ(s). Since the Laurent series is based on the pole at s=1, ζ(s) can be approximated around s=1 using the first two terms of the series: 1/(s-1) + γ. As you get closer to s=1, the higher order terms of the expansion approach 0. I believe that the bigger circular arches created during the transformation are actually created by the 1/(s-1) term, while the constant term shifts the transformed grid to the right by γ.

I was able to confirm this by using manim to render what that approximation of ζ(s) looks like. Those arches look nearly identical to the ones in the Riemann Zeta Function transformation, except the area around the origin looks different. As for why those circular arches are created by vertical lines, I wrote a proof here. A similar method can be used to show how the horizontal lines become circles as well. Note that the radius of the circle created is inversely proportional to the distance of the line Re(s)=a from the pole s=1. So lines closer to s=1 end up creating larger circles after the transformation. I sort of think that the Riemann Zeta Function transformation might be a little misleading because that prominent symmetry is only really caused by behavior near the pole. The more distinct behavior of ζ(s) ends up getting squished to this small area by the origin. I don't think there's really a good way to avoid that though. I am unsure of why exactly f(z)=1/(z-1) seems to be symmetric around the origin though 🤔. Maybe someone could chime in on that.

EDIT: f(z) is a rational function, so I think it would be holomorphic every

https://swgoh.gg/p/197691866

View Poll

In light of the recent post about donating flowers to put on Turing's statue in Manchester on his birthday, I thought I'd share this fun fact about my favorite academic.

A method for the calculation of the zeta-function

I was looking at the formula for the reciprocal of the Riemann Zeta function and its Euler Product Formula here. The argument of the product on the RHS looks very similar to the infinite product of sin(x) used by Euler to solve the Basel problem. I don't know all the details about Weierstrass factorization, but Euler did use that idea of representing a function by multiplying (1-1/(n'th root)) for all the roots of a function. In that sense, the Euler product formula could be interperted as a special function evaluated at 1, with roots at p_n to the power of s like so. This of course has the following special case. Has such a function been studied before? What are some useful properties it may have?

EDIT: I elaborated a bit more in my response to cocompact.

EDIT2: With some help from Prime Obsession by John Derbyshire, I figured out the following series expression for it here
where omega(n) counts the distinct prime factors of n

Fell down the youtube rabbit hole and think I have at least somewhat of an understanding of the RZF's relationship to the primes. Looking at this video, particularly at the end where they overlay a "wave" on top of a "modified prime counting function", it seems like at some point you could just look at where that wave jumps to determine every prime number. I'm sure this faulty logic, I'm just curious where the fault is.

Sidenote, the "wave" they overlay in that video, is that like a fourier series or is that something completely different?

I've seen a number of educational youtube videos about the Reiman Zeta Function. I get the basics for how it works but I'm still a little fuzzy on why it's so important. The video-makers always say something to the effect of "it encodes certain information about prime numbers" but never go into any more detail than that.

So what does it tell us about primes and what are its uses beyond that as well?

Had this thought as i've been playing and it seems like a convenience idea that seems so likely either it exists and I haven't found it, or it's been consider or just hasn't been managed yet. Could there be a way to augment the teleport gun used to go to and from Big MT after OWB with the functionality of Zeta's mothership beacon? For progression's sake it could first require returning to the beacon in DC dropped after MSZ is finished and "uplinking" it somehow with the Big MT transporter, so that using it in either wasteland gives the option to go to either Zeta or Big MT, and when in the latter locations the return function gives an option btw. the Mojave and DC, taking you to the drive-in satellite and beacon, respectively.

So as you all know the zeta function at 2 is pi²/6 But if you use other ways tho calculate this sum You'll find it equal to 0 or 1 or undefined

For example : I tried the ramanujan summation integral

With f(x)=1/x² I got 0

Or the reciprocal of the gamma function at 2

I got 1 and undefined

So what have I done wrong? Or is it the real answer for this sum?

Here is the Riemann Zeta function at x=1/2 in polar form, as u/sargos7 had suggested. I basically parameterized the formulas used for the cartesian representation, which was the more intuitive one for me. The graph appears to happen pretty fast, within a few seconds. I set t from 0 to 34, just like wikipedia did, because after that the graphing seems to take too long and also seems to break.

https://preview.redd.it/bl9tjqhalsu51.png?width=913&format=png&auto=webp&s=465901ec387707d929216fdfe2c361e12b45e670

https://www.desmos.com/calculator/h5kwiqowta?lang=en

https://www.geogebra.org/m/ztquc6xg

Hey all I am an undergraduate at a university that doesn't offer any classes on the Riemann zeta-function. Despite this I want to learn about it, and have been wanting to buy a book and teach myself about it.

I have had classes on:

Linear algebra

Abstract algebra

Some number theory

Introductory calculus(just finished Fourier and series).

In January I will do metric spaces and topology and in the months after I will do real analysis.

Do you folk have any suggestions to literature that would fit my level?

Or maybe I should wait until I know some more?

(I don't mind a challenge, but anything too crazy and I could probably be spending my time more wisely).

Thanks in advance

I mean it must be worth that to somebody if it's a millennium prize but what things that aren't possible now will suddenly be possible as a result?

Hi there, I'm (relatively) new to the Cicada mystery and find it very intriguing. As a mathematician, any puzzle involving prime numbers is highly interesting to me.

As such, I naturally thought of the Riemann Zeta function and its possible connections with the Cicada puzzle. To my surprise, I found no mentions of these two concepts together.

In (very simple) layman's terms, the Riemann Zeta function is a special function in mathematics that is used to describe the distribution of prime numbers. This is one of the most important functions in number theory since it directly connects the zeros ("solutions") of a complex-valued function to the prime numbers.

Now the interesting part is that these "zeros" are all complex - in layman's terms, each "zero" has 2 components in the complex plane: "coordinates" on a map if you will. Perhaps these coordinates somehow relate to the different pages of the Liber Primus?

Has such a connection ever been made?

(again, I apologize for not knowing all the facts, I've just started out with Cicada)

I’m beginning to learn about the Riemann-Zeta function in class and it seems to have so many applications! What are some of the coolest/most interesting ones to you?

Riemann Zeta

Dirichlet Eta

Hello everyone. Here are two domain colorings of functions that take around 5 minutes and 10 minutes, respectively, to appear on my computer due to involving integrations to infinity, but that I decided to upload because I just thought they were cool looking. Here are the Riemann Zeta and the Dirichlet Eta function. As someone may notice from their strange similarity, I've recently learned that they share an intriguing relationship through the polylogarithm function, which may be really interesting to demonstrate if I ever manage to make friends with the daunting scary entity. Anyway, for now I'll just share these two: https://www.desmos.com/calculator/tur3zvbypv?lang=en & https://www.desmos.com/calculator/wo97dahsaf?lang=en, plus maybe this real-valued Dirichlet Eta on real z that appears way way quicker, in just a few seconds https://www.desmos.com/calculator/ah7tcyrbbx?lang=en. As always, I hope you enjoy :)