Two generalizations of Liouville λ function

André Pierro de Camargo
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 1, Pages 30–39
DOI: 10.7546/nntdm.2023.29.1.30-39
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André Pierro de Camargo
Federal University of the ABC region, Brazil

Abstract

We study the properties of two classes of functions \lambda_k and \tilde{\lambda}_k that generalize the Liouville \lambda function, including some equivalencies between the Riemann hypothesis and some assertions about the asymptotic behavior of the summatory functions of \lambda_k and \tilde{\lambda}_k. Similar results are obtained for the generalization of the Möbius function considered by Tanaka.

Keywords

  • Liouville function
  • Möbius function
  • Prime Number Theorem
  • Riemann Hypothesis

2020 Mathematics Subject Classification

  • 11N56
  • 11N99

References

  1. Apostol, T. M. (1976). Introduction to Analytic Number Theory. New York: Springer–Verlag.
  2. Apostol, T. M. (1970). Möbius functions of order 𝑘. Pacific Journal of Mathematics, 32(1), 21–27.
  3. Camargo, A. (2021). Dirichlet matrices: Determinants, permanents and the Factorisatio Numerorum problem. Linear Algebra and Its Applications, 628, 115–129.
  4. Fujisawa, Y. H. (2014). On Mobius and Liouville functions of order k. arXiv:1305.6015v2 [math.NT].
  5. Humphries, P. (2013). The distribution of weighted sums of the Liouville function and Pólya’s conjecture. Journal of Number Theory, 133, 545–582.
  6. Moser, L., & MacLeod. R. A. (1966). The error term for square-free integers. Canadian Mathematical Bulletin, 9(3), 303–306.
  7. Panaitopol, L. (2001). Some properties of Liouville’s function. Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie, 92(4), 365–370.
  8. Pappalardi, F. (2005). A survey on 𝑘-freeness. In: S. D. Adhikari, R. Balasubramanian, & K. Srinivas (Eds.), Number Theory. Lecture Notes Series, Vol. 1, pp. 71–88. Mysore: Ramanujan Mathematical Society.
  9. Tanaka, M. (1980). On Möbius and allied functions. Tokyo Journal of Mathematics, 3(2), 215–218.

Manuscript history

  • Received: 6 September 2022
  • Revised: 15 December 2022
  • Accepted: 10 February 2023
  • Online First: 13 February 2023

Copyright information

Ⓒ 2023 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Camargo, A. P. (2023). Two generalizations of Liouville λ function. Notes on Number Theory and Discrete Mathematics, 29(1), 30-39, DOI: 10.7546/nntdm.2023.29.1.30-39.

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