What is the product of all {\em odious} integers, i.e., of all integers whose binary expansion contains an odd number of $1$'s? Or more precisely, how to define a product of these integers which is not infinite, but still has a "reasonable" definition? We will answer this question by proving that this product is equal to $\pi^{1/4} \sqrt{2 \varphi e^{-\gamma}}$, where $\gamma$ and $\varphi$ are respectively the Euler-Mascheroni and the Flajolet-Martin constants.

中文翻译：

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可恶数的 zeta 正则化乘积

所有 {\em odious} 整数的乘积是多少，即二进制展开包含奇数 $1$ 的所有整数的乘积是多少？或者更准确地说，如何定义这些整数的乘积，它不是无限的，但仍然具有“合理”的定义？我们将通过证明这个乘积等于 $\pi^{1/4} \sqrt{2 \varphi e^{-\gamma}}$ 来回答这个问题，其中 $\gamma$ 和 $\varphi$ 分别是Euler-Mascheroni 常数和 Flajolet-Martin 常数。