论文标题
在一些质量数量的产品上
On some products taken over the prime numbers
论文作者
论文摘要
本文致力于研究$ \ prod_ {p} p^{\ lfloor \ frac {x} {x} {f（p）} \ rfloor} $的一些表达，其中$ x $是一个非负性的实际数字，$ f $是一种满足某些条件的算术功能，并且超过了某些条件，并且超过了primes $ $ $ p $ p。我们首先证明可以通过使用$ \ mathrm {lcm} $函数来表达此类表达，而无需任何引用素数。我们用几个示例说明了这一结果。本文的其余部分专门研究与$ f（m）= m $和$ f（m）= m -1 $有关的两种特殊情况。在这两种情况下，我们都发现了基础表达式的算术特性和分析估计。我们还为案例$ f（m）= m -1 $提出了一个重要的猜想，这取决于特殊表格的质量数的计数。
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.
