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\begin{center}
\vskip 1cm{\Large\bf On Some Products Taken over Prime Numbers
}
\vskip 1cm
\large
Abdelmalek Bedhouche and Bakir Farhi \\
Laboratoire de Math\'ematiques appliqu\'ees \\
Facult\'e des Sciences Exactes \\
Universit\'e de Bejaia \\
06000 Bejaia \\
Algeria \\
\href{mailto:abdelmalek.bedhouche@univ-bejaia.dz}{\tt abdelmalek.bedhouche@univ-bejaia.dz} \\
\href{mailto:bakir.farhi@univ-bejaia.dz}{\tt bakir.farhi@univ-bejaia.dz}
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\begin{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.
\end{abstract}

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