The distribution of consecutive prime biases and sums of sawtooth random variables

TL;DR

This paper investigates biases in the distribution of consecutive prime patterns modulo q, linking these biases to Fourier transforms of Dedekind sums and error terms in number theory, providing new insights into prime pattern frequencies.

Contribution

It introduces a detailed analysis of secondary factors influencing prime pattern biases, connecting them to classical Dedekind sums and error terms, enhancing understanding of prime distribution patterns.

Findings

Biases in prime pattern frequencies are connected to Fourier transforms of Dedekind sums.

The secondary factors in prime biases relate to error terms in summations involving Euler's totient function.

The study offers a new perspective on the distribution of prime patterns modulo q.

Abstract

In recent work, we considered the frequencies of patterns of consecutive primes $\pmod{q}$ and numerically found biases toward certain patterns and against others. We made a conjecture explaining these biases, the dominant factor in which permits an easy description but fails to distinguish many patterns that have seemingly very different frequencies. There was a secondary factor in our conjecture accounting for this additional variation, but it was given only by a complicated expression whose distribution was not easily understood. Here, we study this term, which proves to be connected to both the Fourier transform of classical Dedekind sums and the error term in the asymptotic formula for the sum of $\phi(n)$.