Saturday, July 24, 2010


JSH: Some important info on prime numbers

Some of the arguing about prime numbers that I'm facing may be about not understanding a few basic things about the known and accepted theory on them, so I'll give a quick post to go over that broadly as it is a fascinating subject. I'm going from memory here so if there are errors, sorry, and others may correct.

First off, it was Euler's zeta function.

Mathematicians had noticed that the count of prime numbers aka the prime distribution was related to these continuous functions and Euler found this cool power series that allowed him to find limits. But it was HIS zeta function and NOT Riemann's. Riemann later took Euler's zeta function and did a few things which I'll come back to in a bit.

Chebyshev actually did some great work refining the limits with Euler's zeta function, and his work later became called the prime number theorem for some reason or other, and others have done a lot in that area.

Now Riemann came along later and trying to answer the 'why' of the prime distribution he got creative and pushed Euler's zeta function beyond an asymptotic area with analytic continuation which was a clever way to go into the complex plane into the "imaginary" numbers area.

I've had to think on these areas because of my own prime counting function, which gives a 'why' of the prime distribution by connecting prime counting itself directly to a continuous function through a partial differential equation, which means that my idea could resolve the Riemann Hypothesis directly.

Gauss was interested in primes as well, of course, and did his own work in this area including introducing the Li(x) function. But it wasn't until after he died that twin primes became a big deal and the twin primes conjecture hit the stage in the late 1800's.

There are two things that happen with the twin primes conjecture and prime gaps:
  1. There are empirical formulas based on the actual gaps which tend to include x/ln x or Li(x), which I say is wrapping up the prime distribution with that count, and they often have components that fit with the probabilistic approach I have. They have values that are corrected using actual data, like for a constant called c.

  2. There are bounds given usually through something or other related to the Riemann Hypothesis which are more in the area of derived as ultimately they trace back to Euler's zeta function which has known reasons for being connected to the prime distribution.
My own position is that the probabilistic explanation is THE explanation and that the prime distribution itself is falsely correlated by many math people with the prime gaps as you can just throw it away, and get the answer by considering the number of primes within a particular interval with a probability calculation.

My idea greatly simplifies the area, and dismisses the need for any other explanation, resolving the Twin Primes conjecture with a probabilistic explanation which may be unsatisfying, but worse, the same idea allows you to disprove Goldbach's conjecture which is even more unsatisfying as that proof—or what I think is a proof, kind of hope it's wrong—indicates that a counter-example has such a very low probability that it is unlikely to ever be seen.

Oh, my prime counting function for various reasons also makes it seem likely to me that the Riemann Hypothesis is false, which of course really upsets quite a few apple-carts as established math society opinion has been that it is true.

And that is a quick overview! My research gives a why for the prime distribution, why it is connected to continuous functions like x/ln x, and the why of prime gaps. It also gives a direct path to resolving the Riemann Hypothesis, which I think is false. But it also resolves the Twin Primes Conjecture and Goldbach's Conjecture, but in what is probably an unsatisfying way.

I highly recommend interested readers dig into the history and learn more about Euler's work with his zeta function. Understand how it relates to the prime distribution and learn who Chebyshev was as well as why Gauss came up with Li(x).

It's a fascinating story, and Riemann isn't the only interesting player. But he also was actually working on getting an answer to a problem I answered, so, it's kind of one of those odd things for me talking about him.

Oh, but it IS a fascinating thing to consider that mathematicians might not want the correct answer here, as my research as a way to resolve the Riemann hypothesis has been out there for years.

My suspicion is that for mathematicians whose lives have been turned upside down by my research findings, it's just one more log to toss on the fire which is burning away their research like it was, well, trash, and knowing that I'm probably right is just something they can run away from like they do with the rest of my research.

But why does the world let them? A question for historians, social scientists, and other scholars.

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