Mathematicians will never stop proving the Prime Number Theorem
Interesting fact about the Prime Number Theorum. It is NOT an estimate of the number of primes below a certain N but instead it is an upper bound on the number of primes. The classic formula derived from it, n&#x2F;log(n), is always above the actual number of primes (except for very small &#x27;n&#x27; (&lt;10) where the approximation has significant error).<p>You can compare the actual counts of primes below 10^x (sequence A00688) to the estimates provided by the prime number theorum to see this (sequence A057835) for yourself.<p>Explanation:
At the risk of once again proving the Prime Number Theorum we can use a sieve to illustrate the upper bound. Above 2 there&#x27;s an upper bound of 1&#x2F;2 numbers possible being prime, since only 1&#x2F;2 numbers are not a factor of 2. Above 6 there&#x27;s an upper bound of 1&#x2F;3 numbers being prime (only 1&#x2F;3 numbers above 6 are not a factor of 2 or 3) and the cycle of multiples of 2&amp;3 repeats every multiple of 6 so only 2 number remain that aren&#x27;t a factor of 2 or 3 every multiple of 6. We could say above 30 (which is 2 * 3 * 5) only 8&#x2F;30 numbers are possibly prime since only 8&#x2F;30 numbers are not a factor of 2,3 or 5 on a repeating cycle. We could again create a sieve at 2 * 3 * 5 * 7 (210) with a repeating cycle of those factors (with 48&#x2F;210 numbers remaining that aren&#x27;t factors).<p>The prime number theorum can be derived from this upper bound.<p>If you&#x27;re sieving out factors of a prime each time one is discovered you would be checking 1&#x2F;2 * 2&#x2F;3 * 4&#x2F;5 * 6&#x2F;7 * 10&#x2F;11... etc. numbers for primality. This is essentially the left hand side of the Euler product formula you can see here - <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function#Proof_of_the_Euler_product_formula" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Proof_of_the_Euler_product_for...</a>
Also see <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Euler's_totient_function#Euler&#x27;s_product_formula" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Euler's_totient_function#Eul...</a><p>Now that same totient function is a good way to derive the prime number theorum - <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Prime_number_theorem#Prime_number_theorem_for_arithmetic_progressions" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Prime_number_theorem#Prime_num...</a><p>So the prime number theorum is essentially defining the counts resulting from a sieve of primes. A sieve is an upper bound on the number of primes. Eg. there&#x27;s numbers of the form 6n+1 or 6n+5 that aren&#x27;t prime (eg. 25) that won&#x27;t be filtered by that sieve but also aren&#x27;t prime and as we&#x27;ve seen the prime number theorum can be dericed from this sieve. This is why the prime number theorum is an upper bound that always overestimates the actual number of primes below a certain number.<p>I suspect there are possibly ways to refine the prime number theorum further so that it&#x27;s not always over estimating but i don&#x27;t know how to do this. I can just understand why it&#x27;s always an overestimate.