Note on the cardinality difference between primes and twin primes and its impact on function x/ln(x) in prime number theorem

摘要

We provide a simple explanation and test that the emblematic function x/ln(x) in prime number theorem estimates roughly the prime-counting as 1 for each isolated primes and 1 for each twin primes, and not 2 for twins as it should be. By this reason, for a more proper counting the function (x/ln(x))[1+ln(x)-1] should replace x/ln(x) in the prime number theorem.