Asymptotic expansions for the reciprocal Hardy-Littlewood logarithmic integrals

TL;DR

This paper generalizes existing formulas for generating functions and asymptotic expansions related to prime counting and permutation enumeration, providing insights relevant to the Hardy-Littlewood conjecture.

Contribution

It introduces a new family of recurrences that extend Comtet's and Panaitopol's formulas, leading to generalized asymptotic expansions for prime counting functions.

Findings

Generalized asymptotic expansion for prime counting functions

Extended Comtet's formula for permutation enumeration

Insights into the Hardy-Littlewood conjecture

Abstract

Defining a family of recurrences, we generalize Comtet's formula for the generating function of the enumeration of indecomposable permutations. Consequently, we generalize Panaitopol's asymptotic expansion for the prime counting function, obtaining asymptotic expansions salient to the first Hardy-Littlewood conjecture.