Sunday, November 01, 2009


JSH: Not Riemann's zeta function

It was actually Euler's zeta function and if you don't understand that oddity where math people shifted to saying it was Riemann's zeta function when Euler used it before, then you will not understand a lot of the story!

There is a LOT of history here, so it's a lot easier if you understand the timeline and you actually need to start with Euler.


Euler used the zeta function to first connect the count of prime numbers to a continuous function.


That connection is easy to see by playing with the scientific calculator that should be on your computers.

Count of primes up to 10: 2, 3, 5 and 7. 4 primes 10/ln 10 = 4.3 to one significant digit

Count of primes up to 100: 25 primes. 100/ln 100 = 21.7 to one significant digit

You may not care but for people who were interested in primes it was kind of an odd thing! And Euler FIRST made the connection with a continuous function with his zeta function.

Euler is the one who then found a way to even begin to talk mathematically about how primes actually connected to continuous functions, and was the guy who realized that the zeta function was a way to do so.

The next significant mathematician in the line to Riemann is intriguingly Chebyshev, and not Gauss.


Gauss introduced a function called Li(x), which is closer to the prime count than x/ln x, as notice say, with 1000.

The count of primes to 1000 is 168, but 1000/ln 1000 = 144.7 to one sig no rounding.

So there is this increasing gap between the count of prime numbers and x/ln x, which mathematicians sought to remove!!!

(I don't give values for Li(x) as it's not as easy to just calculate it on your home pc!)

But Chebyshev figured out a way to use Euler's zeta function to put boundaries on the continuous function which became the "prime number theorem". (Math people can get kind of weird with some of their designations.)

Riemann narrowed those boundaries and made his famous hypothesis and assuming it to be true various other mathematicians came up with arguments that should not be called proofs!

They shouldn't be called proofs unless they are established on proof but math people have this annoying habit of calling them proofs and saying they firmly believe the Riemann Hypothesis to be true, without a proof that it IS true.

It's like trying to build a castle without a foundation at all. Building it on air.

But now with ancient history done let's get to modern times.

My research starts you over at the beginning!!!

Back to Euler with the question of how to connect the prime distribution to continuous functions in the FIRST place.

I found easily enough that you have a P(x,y) function, a multi-dimensional prime counter, versus the pi(x) function where mathematicians oddly picked "pi" with the prime counting function which can lead to confusion. It's not pi. It's pi(x), the count of primes up to and including x, so pi(10) = 4. pi(3) = 2, as 2 and 3 are counted.

My P(x,y) function leads to a partial differential equation, thus easily connecting the prime distribution to a continuous function by giving a natural REASON. A simple easy to follow reason for why prime counts would be close to some continuous function.

That is not in debate.

But now it's possible to look over the entire previous history with a new tool and a new perspective as I've given what's likely to be THE explanation, so all others can be evaluated with it.

It's kind of like when Ptolemy's spheres were superseded by Kepler's ellipses.

My research greatly simplifies the area and allows you to now go back and look over what was done previously, including verifying or disproving the Riemann Hypothesis.

However, remember those "proofs" based on the Riemann Hypothesis? If the Riemann Hypothesis is shown now to be false then those arguments are clearly not proofs and a lot of mathematical edifice was shown to be, wrong.

Mathematicians have proudly proclaimed that upheavals that are a part of physics, like the shift from Ptolemy, are not in mathematics which they claim builds on proof.

But with the Riemann story you can see sly cheat throughout, like the shift from the Euler zeta function to the Riemann zeta function, and the building of "proof" on the top of a hypothesis!

Here I can give a prime counting function. Can easily step through to connecting it to a continuous function by getting to a PDE, and it's a multi-dimensional prime counter versus the single variable pi(x) that math people had before, so I can give THE answer and logically connect primes to a continuous function, for the first time in human history.

And I've been able to do that since 2002.

It's weird living history. Years from now scholars can puzzle over the bizarre recalcitrance in mathematicians in fighting what by then will be considered one of the great results in mathematics, just like we can puzzle today about why people would disagree with Galileo.

I mean it's so obvious, right? Clearly the earth revolves around the sun!

It's so obvious here, there's now for the first time a simple answer to the prime distribution!

Belief is weird. Maybe somehow, someway the math people believe they are doing the right thing by fighting the actual mathematical answer to hold on to stuff that has a lot of history, but is probably false, but it's hard to see how.

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