Abstract
The following notation will be used throughout
log r x is the r times iterated logarithm of x,
ψ is Euler’s constant, \(\alpha _0 = \log \Pi _p \;prime\;(1 - \frac{1}{p})^{ - 1/p}\). Density is always the asymptotic density.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
P. T. Bateman, The distribution of values of the Euler function, Acta Arith. 21 (1972), 329–345.
J. Browkin and A. Schinzel, On integers not of the form n-φ(n), Colloq. Math. 68 (1995), 55–58.
H. Davenport, Über numeri abundantes, Sber. Preuß. Akad. Wiss. Berlin, 27 (1933), 830–837, also in the Collected works, vol. 4, 1834–1841.
M. Deléglise, Bounds for the density of abundant integers, Exp. Math. 7 (1998), 137–143.
R. E. Dressier, A density which counts multiplicity, Pacific. J. Math. 34 (1970), 371–378.
P. D. T. A. Elliott, Probabilistic number theory, I Mean-value theorems, Grundlehren der Mathematischen Wissenschaften 239, Springer Verag 1979.
A. Flammenkamp and F. Luca, Infinite families of non-cototients, Colloq. Math. 86 (2000), 37–41.
K. Ford, The distribution of totients, Ramanujan J. 2 (1998), 67–151.
-, The number of solutions of φ(x) = m, Ann. of Math. 150 (1999), 283–311.
-, An explicit sieve bound and small values of σ(φ(m)), Period. Mat. Hungar, 43 (2011), 15–29.
K. Ford, F. Luca and C. Pomerance, Common values of the arithmetic functions φ and σ, Bull. London Math. Soc. 42 (2010), 478–488.
K. Ford and P. Pollack, On common values of ϕ(n) and σ(n), I, Acta Math. Hungar. 133 (2011), 251–271.
S. W. Graham, J. J. Holt and C. Pomerance, On the solutions to ϕ(n) = ϕ(n +k), Number theory in Progress, vol. 2, 867–882, Walter de Gruyter 1999.
P. Loomis and F. Luca, On totient abundant numbers, Integers. Electronic J. of Combinatorial Number Theory 8 (2008), #A06.
A. Ivić, The distribution of positive abundant numbers, Studia Sc. Math. Hungar. 20 (1985), 183–187.
F. Luca and C. Pomerance, On some problems of Mqkowski-Schinzel and Erdös concerning the arithmetical functions φ and σ, Colloq. Math. 92 (2002), 111–130; Acknowledgement of priority, ibid. 126 (2012), 139.
—, On the range of the iterated Euler function, Combinatorial number theory, 101–106, Walter de Gruyter, Berlin, 2009.
H. Maier, On the third iterates of the φ-and σ-function, Colloq. Math. 49 (1984), 123–130.
S. S. Pillai, On some functions connected with φ(n), Bull. Amer. Math. Soc. 35 (1929), 832–836.
P. Pollack, Long gaps between deficient numbers, Acta Arith. 146 (2011), 33–43.
-, Two remarks on iterates of Euler’s totient function, Arch. Math. 97 (2011), 449–453.
-, On the greatest common divisor of a number and its sum of divisors, Michigan Math. J. 60 (2011), 199–214.
C. Pomerance, On the distribution of amicable numbers, J. Reine Angew Math. 293/294 (1977), 217–222.
On the composition of the arithmetic functions φ and ϕ, Colloq. Math. 58 (1989), 11–15.
P. Poulet, Nouvelles suites arithmétiques, Sphinx 2 (1932), 53–54.
A. Schinzel, On functions φ(n) and σ(n), Bull. Acad. Polon. Sci. Cl. III, 3 (1955), 415–419.
E. Wirsing, Bemerkung zu der Arbeit über volkommene Zahlen, Math. Ann. 137 (1959), 316–318.
K. Wooldridge, Values taken many times by Euler’s phi-function, Proc. Amer. Math. Soc. 76 (1979), 229–234.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 János Bolyai Mathematical Society and Springer-Verlag
About this chapter
Cite this chapter
Schinzel, A. (2013). Erdős’s Work on the Sum of Divisors Function and on Euler’s Function. In: Lovász, L., Ruzsa, I.Z., Sós, V.T. (eds) Erdős Centennial. Bolyai Society Mathematical Studies, vol 25. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39286-3_21
Download citation
DOI: https://doi.org/10.1007/978-3-642-39286-3_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-39285-6
Online ISBN: 978-3-642-39286-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)