Correlations of error terms for weighted prime counting functions

TL;DR

This paper investigates the correlations between error terms of prime counting functions and their weighted variants, revealing links to the Riemann hypothesis and calculating densities of sign coincidences under certain assumptions.

Contribution

It analyzes correlations among error terms of prime counting functions, connecting persistent inequalities to the Riemann hypothesis and computing sign density probabilities.

Findings

Persistent inequalities between error terms are equivalent to RH.

Conditional densities of error term sign coincidences are approximately 98.65%.

Error terms of different prime counting functions share the same sign with high probability.

Abstract

Standard prime-number counting functions, such as $\psi(x)$, $\theta(x)$, and $\pi(x)$, have error terms with limiting logarithmic distributions once suitably normalized. The same is true of weighted versions of those sums, like $\pi_r(x) = \sum_{p\le x} \frac1p$ and $\pi_\ell(x) = \sum_{p\le x} \log(1-\frac1p)^{-1}$, that were studied by Mertens. These limiting distributions are all identical, but passing to the limit loses information about how these error terms are correlated with one another. In this paper, we examine these correlations, showing, for example, that persistent inequalities between certain pairs of normalized error terms are equivalent to the Riemann hypothesis (RH). Assuming both RH and LI, the linear independence of the positive imaginary parts of the zeros of $\zeta(s)$, we calculate the logarithmic densities of the set of real numbers for which two different error terms have prescribed signs. For example, we conditionally show that $\psi(x) - x$ and $\sum_{n\le x} \frac{\Lambda(n)}n - (\log x - C_0)$ have the same sign on a set of logarithmic density $\approx 0.9865$.