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

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Abstract

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 \(\varPi (x)\).

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Authors and Affiliations

  1. Federal University of the ABC Region, Santo André, Brazil

    André Pierro de Camargo

  2. Institute of Mathematics and Statistics of University of Sao Paulo, São Paulo, Brazil

    Paulo Agozzini Martin

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  1. André Pierro de Camargo
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  2. Paulo Agozzini Martin
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Correspondence to André Pierro de Camargo.

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Camargo, A.P.d., Martin, P.A. Constant Components of the Mertens Function and Its Connections with Tschebyschef’s Theory for Counting Prime Numbers. Bull Braz Math Soc, New Series 53, 501–522 (2022). https://doi.org/10.1007/s00574-021-00267-4

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  • DOI: https://doi.org/10.1007/s00574-021-00267-4

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