Prime Running Functions

Abstract

We study arithmetic functions $\Phi(x;d,a)$, called prime running functions, whose value at $x$ sums the gaps between primes $p_k \equiv a\ (\text{mod}\ d)$ below $x$ and the next following prime $p_{k+1}$, up to $x$. (The following prime $p_{k+1}$ may be in any residue class $(\text{mod}\ d)$.) We empirically observe systematic biases of order $x / \log x$ in $\Phi(x;d,a) - \Phi(x;d,b)$ for different $a,b$. We formulate modified Cram\'er models for primes and show that the corresponding sum of prime gap statistics exhibits systematic biases of this order of magnitude. The predictions of such modified Cram\'er models are compared with the experimental data.