Wednesday, August 16, 2006

 

JSH: Why Riemann was likely wrong

I posted a while back that the Riemann Hypothesis was probably wrong, as I'm still not ready to say that I've proven it wrong.

Here's why it's probably wrong.

I noted that Mertens's theorem can be related to prime probability as revealed by a simple idea, and it turns out that the prime number theorem actually traces back for its most important results to only two people: Euler and Chebyshev.

And it is the Euler zeta function.

Mathematicians have progressively written Euler out of the picture till today I'm sure there are young people who would proudly proclaim it Riemann's zeta function, when nope, it's Euler's.

Euler figured out the power series and Chebyshev did some really nice work figuring out the limits which are the prime number theorem.

That is the most important work done in this area.

What Riemann did was try to look back into an imaginary part of the Euler zeta function, and it just logically doesn't make sense that that part has any applicability, especially when you understand the prime probability explanation and why Euler's and Chebyshev's work apply in the first place.

Any of you know about the Mandelbrot set?

You should. And you should know that within the Mandelbrot set there are similar versions of the larger set, all the way to infinity.

I suggest to you that to any extent that the imaginary portion found by analytic continuation of Euler's zeta function appears to match the prime counting function may just be the same kind of thing, where you have something similar to the main set, but not exact.

Besides, if you read Riemann's own words—I read a translation from the German—his approach doesn't sound right as he talks about hoping some stuff balances out over infinity.

Mathematics doesn't work that way. Either things do or they don't, and if they do, it can be proven. No guesswork required.

It's probably not hard to prove it definitively given the connection I've shown with my prime counting function as well as this latest research with prime probability as I've sewn up prime behavior.

But the politics in this area are rather heavy, so I suspect there will still be some mud-slinging from both sides for a while yet.

But for the more intelligent of you, who actually care about what is true, my assessment based on the evidence in front of me now is that the Riemann Hypothesis is false.





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