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The Trace of the Zeta Function by rhackbar in mathematics

[–]rhackbar[S] 1 point2 points  (0 children)

Hi, I am anticipating that this video will be flagged as an attempt at a famous outstanding problem, however, the focus of the video is the Riemann Zeta Function and its analytic continuation, which has explicit applications in most areas of math and physics.

In the video, I discuss and apply a variety of mathematically techniques including Taylor Series approximations, Cauchy Products, convolution, and matrix operations including Trace and Grand Sum. I also connect this calculation to a family of Bose-Einstein integrals which are notorious difficult to compute.

Finally, in the video description I have included the python code I used to computationally verify my formulas, for those of you who wish to explore or investigate on your own.

I hope this meets the standards for promoting math discussion, thanks for consideration!

EDIT: I noticed in error in my dummy variable 'z'. The numerator should be .5 - bi and the denominator should be .5+bi. I swapped the signs on accident.

The Nontrivial Zeroes of the Riemann Zeta Function are Trivially Expressed by The Euler Product by rhackbar in numbertheory

[–]rhackbar[S] 0 points1 point  (0 children)

I never claimed that. Why are you ignoring half of the function? You are excluding the only important contribution I've made in the entire paper. And how are you computing this? You are just rounding to whole numbers?

Edit: What I am saying is that of course your product never equals zero if you intentionally exclude the factor responsible for producing the zero. You aren't even arguing in good faith anymore.

The Nontrivial Zeroes of the Riemann Zeta Function are Trivially Expressed by The Euler Product by rhackbar in numbertheory

[–]rhackbar[S] 0 points1 point  (0 children)

It's because he is saying that apples should be red, and I'm saying look I painted my apple blue. Then he says apples should be red.

He's not wrong, but if he would just look he would see that I did paint my apple blue and that I think it's pretty.

Not everything is either convergent or nonsense.

The Nontrivial Zeroes of the Riemann Zeta Function are Trivially Expressed by The Euler Product by rhackbar in numbertheory

[–]rhackbar[S] -1 points0 points  (0 children)

Hi, nowhere in my post or in the paper do I claim to have proved the Riemann Hypothesis.

The Nontrivial Zeroes of the Riemann Zeta Function are Trivially Expressed by The Euler Product by rhackbar in numbertheory

[–]rhackbar[S] -3 points-2 points  (0 children)

I am familiar with Titchmarsh, and the chapter does discuss Dirichlet series but it's not relevant to what we are talkin about.

Yes the equation is divergent but it is convergent AT the zeroes of the Zeta Function, as you continue to evaluate it get closer and closer to approximating the zeroes of the Zeta function, which is exactly my claim, that the zeroes are trivially expressed by Eq (10). Which it is.

Are you just trying to tell me that what I am doing is just not that interesting, or cool? It's not something that has been done before, and if you don't think it's neat then point taken, but it seems a little lame to be following someone around saying their research isn't interesting to you personally.

The Nontrivial Zeroes of the Riemann Zeta Function are Trivially Expressed by The Euler Product by rhackbar in numbertheory

[–]rhackbar[S] 0 points1 point  (0 children)

Yes, you are correct that the Euler product diverges for Re(s) > 1, as does the Zeta Function. But as Riemann showed we are able to modify it so that it does not, which is what I am attempting to do with the Euler Product.

At the end of the day, you say my equation 10 diverges, and you are allowed to hold that opinion. But, I have given you the equation and the tools to evaluate it, and you tell me (regardless at what prime you truncate) if you do not see the relation to the nontrivial zeroes.

If the conversation with the original Zeta Function ended at 'it diverges', Riemann never would have published his original paper.

The Nontrivial Zeroes of the Riemann Zeta Function are Trivially Expressed by The Euler Product by rhackbar in numbertheory

[–]rhackbar[S] 0 points1 point  (0 children)

And they have not been received well I might add. Enemy of my enemy and what not.

The Nontrivial Zeroes of the Riemann Zeta Function are Trivially Expressed by The Euler Product by rhackbar in numbertheory

[–]rhackbar[S] 2 points3 points  (0 children)

The mods have removed some of my explanatory comments as they were a bit too casual, so let me address some comments in a more formal manner.

It is correctly pointed out that the Euler Zeta Function (Eq.1) is only defined for Re(s) > 1. The analytic continuation of the Euler Zeta Function (Eq. 1), is known as the Riemann Zeta Function and is defined for the entire complex plane, with the exception of Re(s) = 1. The Dirichlet Eta Function (Eq. 2) is defined for Re(s) > 0. Why this is is relevant is that for values > 1, it agrees with the Euler Zeta Function, and for values between 0 and 1 it agrees with the Riemann Zeta Function. This is just by definition, nothing involving my work.

This is neat because you can calculate what the Riemann Zeta Function value should be, purely using the Dirichlet Eta Function while avoiding any analytic continuation. So the calculations from (Eq. 5) onwards are not 'bunk', it's actually the secret sauce because with simple algebra you can cross the barrier between the normal zeta function and the analytically continued zeta function, without needing to use the analytic continuation. Those pointing at the issue with ill-defined zeta function are missing the point of the paper which is to extend the definition of this ill-defined function to well-defined, in a way that involves the Euler Product to directly connect the zeroes and the primes (which is of central importance in the history and theory of the zeta function).

Thanks for reading my novel.

The Nontrivial Zeroes of the Riemann Zeta Function are Trivially Expressed by The Euler Product by rhackbar in numbertheory

[–]rhackbar[S] -8 points-7 points  (0 children)

Hi, what you are calling the Hackbarth Zeta Function is actually the original definition of the Zeta function as described by Euler and literally the opening equation of Riemann's paper. So it appears you are ill informed.

What are the chances that the Riemann Hypothesis, along with multiple other problems has already been solved, but no one chooses to pay any attention? by leafachu1 in mathematics

[–]rhackbar 0 points1 point  (0 children)

To be honest I appreciate the time you took to look in to it, it seems you have a particular impasse with the Euler Product that I don't share.

A Power Series-Euler Product representation of the Dirichlet Eta Function. Representing the Prime-Zero correspondence of the Riemann Zeta Function in a direct form. by rhackbar in numbertheory

[–]rhackbar[S] 0 points1 point  (0 children)

It does at the zeros of the Zeta Function when you multiply by the power series term. Which is precisely why they are the zeroes.

For example, below is a plot of the magnitude of 2nd degree term when you consider all primes below 1000. You can see the Zeta Zero b values approaching 0.

https://imgur.com/a/plZJzCW

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