Journal of Integer Sequences, Vol. 26 (2023), Article 23.1.5

On Some Products Taken over Prime Numbers


Abdelmalek Bedhouche
Laboratoire des Mathématiques Appliquéees
Faculté des Sciences Exactes et Informatique
Université de Jijel
18000 Jijel
Algeria

Bakir Farhi
Laboratoire de Mathématiques Appliquées
Faculté des Sciences Exactes
Université de Bejaia
06000 Bejaia
Algeria

Abstract:

We study 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 reference to prime numbers; we illustrate this result with several examples. The rest of the paper is devoted to studying two particular cases related to $f(m) = m$ and $f(m) = m - 1$. In both cases, we find arithmetic properties and analytic estimates for the underlying expressions.

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(Concerned with sequences A048803 A091137.)

Received July 26 2022; revised versions received November 1 2022; December 15 2022. Published in Journal of Integer Sequences, January 2 2023.

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