Harristotelian Logic

I talk about two forms with my prime counting function, where a lot of discussion ends up about the sieve form.

Here's the sieve form:

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))}

except when n is greater than the number of primes up to sqrt(x), as if it is, it is reset to that count.

It's a sieve form because p_i is the i_th prime number. For instance 3 is the second prime number so

p_2 = 3

and p_3 = 5, as 5 is the third prime number.

So if you want to count prime numbers with the sieve form, you have have a list of primes up to the positive value of sqrt(x), like to calculate P(100,4), you first need to know the first four primes: 2, 3, 5 and 7.

To calculate P(1000000,168) you need to know the first 168 primes.

The other form is NOT a sieve and relies on a partial difference
equation:

With all natural numbers:

P(x,y) = x - 1 - sum for k=2 to y of {(P([x/k,k-1) - P(k-1,sqrt(k-1)))dP(k,sqrt(k)}

where dP(k,sqrt(k)) = P(k,sqrt(k)) - P(k,sqrt(k-1))

where if y is greater than sqrt(x) then y is reset to sqrt(x).

The sieve form can be related to past known research with counting primes:

P(x,n) = phi(x,n) + n - 1

as has been pointed out, but to my knowledge, no one wrote it that way.

Mathematicians always wrote the prime counting function as

pi(x) = phi(x,n) + n -1

as they saw it as a single variable function that had a multi-variable helper function, which is a sieve function.

They didn't make the leap to looking at it as a multi-variable function as there didn't seem to be a point with the prime count.

But I did things my own way, as I did all my discovery from scratch, not having bothered to read about what mathematicians had done on prime counting and I naturally went to a multi-variable prime counting function that is not a sieve function, and can't be directly related to past research unless you force it to a sieve function.

So the natural form for what I discovered is NOT a sieve, it does not need you to give it a list of primes, and it relies on summing a partial difference equation, with a special constraint on how you do that sum.

That form leads to a partial differential equation and you can integrate that to get an approximate count of prime numbers explaining the prime number theorem.

But in so doing you can see that the exact count is different from those results because of that requirement about cutting y off at sqrt(x), so you have this simple explanation for the 'why' of the prime number theorem, as to why there is a gap.

Mathematicians since Gauss were curious about that gap, and Riemann was looking for an explanation when he made his famous hypothesis.

My research gives a simpler explanation and may give a way to check the Riemann Hypothesis that mathematicians did not realize existed, by directly comparing with the function that results from the partial differential equation that follows from the partial difference equation of my prime counting function.