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On Some Products Taken over Prime Numbers Abdelmalek Bedhouche and Bakir Farhi
Laboratoire de Mathématiques appliquées
Faculté des Sciences Exactes
Université de Bejaia
06000 Bejaia
Algeria
abdelmalek.bedhouche@univ-bejaia.dz
bakir.farhi@univ-bejaia.dz

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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.