Constant Components of the Mertens Function and Its Connections with Tschebyschef’s Theory for Counting Prime Numbers II

In our previous article (Camargo and Martin in Bull Braz Math Soc New Ser 53:501–522, 2022), we presented some families of sets \(\varTheta _x \subset \{1, 2, \dots , \lfloor x \rfloor \}\) ${\Theta }_x\subset \{1,2,\cdots ,\lfloor x\rfloor \}$ such that the sum of the Möbius function over \(\varTheta _x\) ${\Theta }_x$ is constant and equals to \(-1\) $-1$ and we showed that the existence of such sets is intimately connected with the existence of the alternating series used by Tschebyschef and Sylvester to bound the prime counter function \(\varPi (x)\) $\Pi (x)$ . In this note, we answer two open questions stated in the last section of (Camargo and Martin 2022) about the general structure of these constant functions. In particular, we show that every such constant function \(x \longmapsto \sum \nolimits _{j \ \in \ \varTheta _x} \mu (j)\) $x⟼{\sum }_{j\in {\Theta }_x}\mu (j)$ can be characterized by Tschebyschef–Sylvester alternating series. We also show that the asymptotic sizes of the sets \(\varTheta _x\) ${\Theta }_x$ connects to the Sylvester’s Stigmata of the Tschebyschef–Sylvester series.