I'm afraid there are quite a lot of errors here. (Not everything in the following is pointing out errors, but most of it is.) 1. Yes, the quantity in the PNT i

I'm afraid there are quite a lot of errors here. (Not everything in the following is pointing out errors, but most of it is.) 1. Yes, the quantity in the PNT is an estimate of the number of primes below a certain N. As usually stated among mathematicians, PNT is the statement that the number of primes up to x is asymptotic to Li(x), the logarithmic integral, and this is known to be sometimes over and sometimes under. 2. If you use x / log x instead of Li(x), then indeed it is known that once x isn't very small this is always an overestimate. But, again, PNT isn't the statement that it's an overestimate, it's the statement that it's a pretty good estimate. If Chebyshev or someone of the sort had proved that x / log x is an overestimate for the number of primes up to x, it would have been hailed as a fine achievement but not as a proof of PNT. 3. But your argument about "sieving out factors of a prime each time one is discovered" is not anywhere near to being a proof of PNT, let alone the stronger theorem that x / log x is an overestimate (which requires some concrete bounds on the error term in the approximation pi(x) ~ Li(x); the way you prove that x / log x is an overestimate is to show that x / log x is bigger than Li(x) by a certain amount, and then show that the difference between pi(x) and Li(x) is smaller than that certain amount; the latter is a strengthened version of the Prime Number Theorem). 4. The Wikipedia link you give is not saying that the Euler totient function is "a good way to derive the prime number theorem". It is stating a nice generalization of the prime number theorem that has the Euler totient function in its statement. 5. When you say "the prime number theorem is essentially defining the counts resulting from a sieve of primes. A sieve is an upper bound on the number of primes. [...] This is why the prime number theorem is an upper bound", I'm afraid this is completely wrong. If there is a proof of the prime number theorem that amounts to saying "look at this sieve, which easily gives an upper bound that's easily seen to be x / log x", then I don't believe anyone's found it yet. 6. It's "theorem", not "theorum".