Abstract
We study the local behavior of the composition of the aliquot function \(s(n)=\sigma (n)-n\) and the co-totient function \(s_{\varphi }(n)=n-\varphi (n)\), where \(\sigma \) is the sum-of-divisors function and \(\varphi \) is the Euler function. In particular, we show that \(s\mathbin {\mathchoice{{\scriptstyle \circ }}{{\scriptstyle \circ }}{{\scriptscriptstyle \circ }}{{\scriptscriptstyle \circ }}}s_{\varphi }\) and \(s_{\varphi }\mathbin {\mathchoice{{\scriptstyle \circ }}{{\scriptstyle \circ }}{{\scriptscriptstyle \circ }}{{\scriptscriptstyle \circ }}}s\) are independent in the sense of Erdős, Győry, and Papp.
For Krishna Alladi on his 60th birthday
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
A. Balog, C. Pomerance, The distribution of smooth numbers in arithmetic progressions. Proc. Amer. Math. Soc. 115, 33–43 (1992)
N. Doyon, F. Luca, On the local behavior of the Carmichael \(\lambda \)-function. Mich. Math. J. 54, 283–300 (2006)
P. Erdős, On a problem of G. Golomb. J. Austral. Math. Soc. 2, 1–8 (1961–1962)
P. Erdős, A. Granville, C. Pomerance, C. Spiro, On the normal behaviour of the iterates of some arithmetic functions, in Analytic Number Theory, Proc. Conf. in honor of Paul T. Bateman, eds. by B. C. Berndt, et al. (Birkhauser, Boston, 1990), pp. 165–204
P. Erdős, K. Győry, Z. Papp, On some new properties of functions \(\sigma (n), \phi (n), d(n)\) and \(\nu (n)\). Mat. Lapok 28, 125–131 (1980)
F. Luca, C. Pomerance, On some problems of Makowski-Schinzel and Erdős concerning the arithmetical functions \(\phi \) and \(\sigma \). Colloq. Math. 92, 111–130 (2002)
F. Luca, C. Pomerance, The range of the sum-of-proper-divisors function. Acta Arith. 168, 187–199 (2015)
M.-O. Hernane, F. Luca, On the independence of \(\sigma (\phi (n))\) and \(\phi (\sigma (n))\). Acta Arith. 138, 337–346 (2009)
H.L. Montgomery, R.C. Vaughan, The exceptional set in Goldbach’s problem. Acta Arith. 27, 353–370 (1975)
J. Pintz, Recent results on the Goldbach conjecture, in: Elementare und analytische Zahlentheorie, vol. 20, Schr. Wiss. Ges. Johann Wolfgang Goethe Univ. Frankfurt am Main, (Franz Steiner Verlag, Stuttgar, 2006), pp. 220–254
P. Pollack, C. Pomerance, Some problems of Erdős on the sum-of-divisors function. Trans. Am. Math. Soc. Series B 3, 1–26 (2016). https://doi.org/10.1090/btran10
C. Pomerance, On the distribution of amicable numbers. J. Reine Angew. Math. 293/294, 217–222 (1977)
H.N. Shapiro, Addition of functions in probabilistic number theory. Commun. Pure Appl. Math. 26, 55–84 (1973)
T. Xylouris, Über die Nullstellen der Dirichletschen \(L\)-Funktionen und die kleinste Primzahl ineiner arithmetischen Progression, Doctoral dissertation, Universiteit Bonn, 2011
N. Lebowitz-Lockard, P. Pollack. J. Number. Theor. 180, 660–672 (2017)
Acknowledgements
This paper was written when F. L. visited Dartmouth College in Summer 2015. He thanks the Mathematics Department there for its hospitality and support. We thank Paul Pollack for suggesting the relevance of [13]. We also thank the referee for a very careful reading.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Luca, F., Pomerance, C. (2017). Local Behavior of the Composition of the Aliquot and Co-Totient Functions. In: Andrews, G., Garvan, F. (eds) Analytic Number Theory, Modular Forms and q-Hypergeometric Series. ALLADI60 2016. Springer Proceedings in Mathematics & Statistics, vol 221. Springer, Cham. https://doi.org/10.1007/978-3-319-68376-8_27
Download citation
DOI: https://doi.org/10.1007/978-3-319-68376-8_27
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-68375-1
Online ISBN: 978-3-319-68376-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)