Prime Running Functions

TL;DR

This paper investigates prime running functions that sum prime gaps in specific residue classes, revealing systematic biases and comparing empirical data with modified Cramér models to understand these biases.

Contribution

It introduces modified Cramér models to explain observed biases in prime gap sums across residue classes, supported by empirical data analysis.

Findings

Systematic biases of order x / log x in prime running functions.

Modified Cramér models predict these biases accurately.

Empirical data confirms the models' predictions.

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.