In this note we exhibit some large sets \(\varTheta _x \subset \{1, 2, \ldots , \lfloor x \rfloor \}\) such that the sum of the Möbius function over \(\varTheta _x\) is small and independent of x. We show that the existence of some of these sets are intimately connected with the existence of the alternating series used by Tschebyschef and Sylvester to bound the prime counter function \(\varPi (x)\).

中文翻译：

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Mertens 函数的常数分量及其与 Tschebyschef 素数计数理论的联系

在这篇笔记中，我们展示了一些大集合\(\varTheta _x \subset \{1, 2, \ldots , \lfloor x \rfloor \}\)使得莫比乌斯函数在\(\varTheta _x\)上的总和是小且独立于x。我们表明，其中一些集合的存在与 Tschebyschef 和 Sylvester 用来限制质数计数器函数\(\varPi (x)\)的交替级数的存在密切相关。