Asymptotics on a class of Legendre formulas

TL;DR

This paper introduces a generalized factorial concept linked to a positive function over primes, demonstrating its asymptotic behavior and providing approximations for related factorials like the Bhargava factorial.

Contribution

It constructs a new class of factorials based on a positive prime function and establishes their asymptotic properties, extending classical Stirling approximations.

Findings

The $f$-factorial satisfies a weak Stirling approximation.

Provides weak asymptotic formulas for the Bhargava factorial.

Extends Legendre formulas to a broader class of functions.

Abstract

Let $f$ be a real-valued function of a single variable such that it is positive over the primes. In this article, we construct a factorial, $n!_f$, associated to $f$, called the associated Legendre formula, or $f$-factorial, and show, subject to certain criteria, that $n!_f$ satisfies a weak Stirling approximation. As an application, we will give weak approximations to the Bhargava factorial over the set of primes and to a less well-known Legendre formula.