Harristotelian Logic

One intriguing line that has come up recently with my prime counting research is the assertion that it is well-known.

However, I JUST did a survey of math websites doing a search on "prime counting" and saw nothing like the following concise prime counting function, which is mine.

My prime counting function in its sieve form is as follows.

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.

That is very concise, and it is in one line where you have the summation with my prime counting function in sieve form recursively calling itself.

It has been noted that it can be directly related to the phi function which usually is discussed with Legendre's Method.

But, um, I can see nothing out there on the web that gives anything like my prime counting function, so, what if?

What if there isn't anything on the web because no one figured it out before?

I figured it out along with posters on the newsgroups years ago because of all those arguments about my research, where it was worked out that it could be directly related to the phi function.

But what if the math world itself never knew until recently?

Questions that can be answered by links or references.

I am asking posters who claim that a concise way to represent the phi function was known previous to my research to give some links or references of some sort.

I am actually curious at this point. I doubt that MathWorld and websites like it would ignore a very succinct way to write the prime counting function if they knew about it before me, or that they would ignore simple optimizations that make counting primes faster, if they knew about them before me.

Yeah, maybe math people hate short and simple which is why you can STILL at this time as I just checked not find anything like what I have on MathWorld.

Is that satisfying for you? Believe they know this information but just hate short and simple?

I don't think so.

But I am quite confident that they would ignore anything and everything if it followed from my research.