The origin of the logarithmic integral in the prime number theorem

Abstract

We establish why li(x) outperforms x/log x as an estimate for the prime counting function pi(x). The result follows from subdividing the natural numbers into the intervals s_k :={p_k^2,..., p_{k+1}^2-1}, k>=1, each being fully sieved by the k first primes {p_1,..., p_k}. Denoting the number of primes in s_k by pi_k, we show that pi_k |s_k|/log p_{k+1}^2 and that pi(x) li(x) originates as a continuum approximation of the sum sum_k pi_k. In contrast, pi(x) x/log x stems from sieving repeatedly in regions already completed---explaining why x/log x underestimates pi(x). The explanatory potential arising from defining s_k appears promising, evidenced in the last section where we outline further research.