Statistical and structural analysis of the appearance of prime numbers

Abstract

We examine the distribution of the ratio of addition to multiplication over standard atomic sets of integers. By analyzing the array of conversion ratios and selected sub-arrays, we prove that the reciprocal of the mean of the conversion ratio distribution converges to the prime-counting function π(n). We also show that the modified mean of the sub-array C ⁵, which is obtained from the array of conversion ratios by scaling and translation, converges to π(n) with an accuracy comparable to the Li-function. We go on to numerically show that the relative behaviors of L(n), $1/H_{n}^{5}$ and Li(n) with respect to π(n) are similar, and that π(n)/L(n), $\pi(n)/(1/H_{n}^{5})$ and π(n)/Li(n) provide approximations of competitive accuracy at the center of the distribution.