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arXiv · 2207.07957

On some products taken over the prime numbers

Abdelmalek Bedhouche·Bakir Farhi
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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.

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Abdelmalek Bedhouche, Bakir Farhi. 2022-07-16. On some products taken over the prime numbers. https://arxiv.org/abs/2207.07957

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