Irregularities in the distribution of primes and twin primes

Abstract:

The maxima and minima of $\langle L(x)\rangle - \pi (x),\langle R(x)\rangle - \pi (x)$, and $\langle {L_2}(x)\rangle - {\pi _2}(x)$ in various intervals up to $x = 8 \times {10^{10}}$ are tabulated. Here $\pi (x)$ and ${\pi _2}(x)$ are respectively the number of primes and twin primes not exceeding $x,L(x)$ is the logarithmic integral, $R(x)$ is Riemann’s approximation to $\pi (x)$, and ${L_2}(x)$ is the Hardy-Littlewood approximation to ${\pi _2}(x)$. The computation of the sum of inverses of twin primes less than $8 \times {10^{10}}$ gives a probable value $1.9021604 \pm 5 \times {10^{ - 7}}$ for Brun’s constant.