On some products taken over the prime numbers

TL;DR

This paper investigates products over primes involving a function of primes, expressing them via the least common multiple, and explores specific cases with arithmetic properties, estimates, and a conjecture related to prime numbers of special forms.

Contribution

It introduces a prime-free expression for products over primes involving an arithmetic function and studies particular cases with new properties and conjectures.

Findings

Expressed prime products using the lcm function without prime references

Derived arithmetic properties and estimates for specific cases

Proposed a conjecture related to primes of special forms

Abstract

This paper is devoted to study some expressions of the type $\prod_{p} p^{\lfloor\frac{x}{f(p)}\rfloor}$, where $x$ is a nonnegative real number, $f$ is an arithmetic function satisfying some conditions, and the product is over the primes $p$. We begin by proving that such expressions can be expressed by using the $\mathrm{lcm}$ function, without any reference to prime numbers; we illustrate this result with several examples. The rest of the paper is devoted to study the two particular cases related to $f(m) = m$ and $f(m) = m - 1$. In both cases, we found arithmetic properties and analytic estimates for the underlying expressions. We also put forward an important conjecture for the case $f(m) = m - 1$, which depends on the counting of the prime numbers of a special form.