Harristotelian Logic

I've repeatedly emphasized the simple sieve form of my prime counting function to make a point that it is new, and unique in ways clearly seen to what was known before as you can just do a web search on "prime counting function" and easily see what math people already knew and find nothing close to the following.

With natural numbers x and n, where p_i is the i_th prime:

P(x,n) = x - 1 - sum for i=1 to n of {(P(x/p_i,i-1) - (i-1))}

where if n is greater than the count of primes up to and including sqrt(x) then n is reset to that count.

I don't think you can find another multi-variable prime counting function in the math literature at all. I know I haven't found one, and it's another crucial point to make against people posting against this research.

But now things get a bit more complicated and I worry about making a more complicated post, but I need to address more directly just how important this research must be, and for that, there will need to be some calculus.

Because my prime counting function recursively call itself you can go from the sieve form to a fully mathematicized form with the following.

With natural numbers x and y, if y<sqrt(x) then

P(x,y) = floor(x) - 1 - sum for k=2 to y of (P(x/k,k-1) - P(k-1,sqrt(k-1)))( P(k,sqrt(k) - P(k-1,sqrt(k-1))))

else P(x,y) = P(x,sqrt(x)).

That may look a lot more complicated but the main difference here is that because

P(k,sqrt(k) - P(k-1,sqrt(k-1))

will equal 0 if k is not prime, I can use the prime counting function itself as a switch which will zero out everything if k is not prime, so you only get non-zero values when k is prime, just like before.

And now you have something never before seen at all with a prime counting function, which is a difference equation as part of it, and more specifically, what is being summed is a partial difference equation.

Not surprisingly, you can go from that to a partial differential equation:

P'y(x,y) = -(P(x/y,y) - P(y, sqrt(y))) P'(y, sqrt(y))

and again find something never before seen with a prime counting function.

So now to believe that my research is actually old you have to believe that no one bothered to go to a partial differential equation from the earlier form.

In its sieve form my prime counting function can be related directly to other sieve prime counting functions and elements within them.

Numerical integration of the partial differential equation can reveal if it is close to the prime count, and if it is, then it stands to reason also that it would be related to continuous functions that are close to that count as well, and more specifically if it is very close, to any functions that follow from the Riemann Hypothesis.

The other possibility is that despite following directly from the prime counting function, the numerical integration of the partial differential equation is completely unrelated to the prime count.

However, my own attempts at doing that integration show it to be closer than Li(x) but a bit further from the prime count than R(x), the Riemann function.

If Legendres had my prime counting function but failed to figure out the fully mathematicized form and failed to figure out the partial differential equation, then he missed some obvious things, and not only him but mathematicians that followed him did as well, including Riemann himself.

Proper follow through on this research alone should make headlines around the world.

It is just some of my number theory research.

For instance, I have also proven Fermat's Last Theorem.