Explicit upper bounds for some special arithmetic functions

The following bounds may be useful is some applications. From [Robin, 1983a †Robin, G. 1983a
Estimation de la fonction de Tchebychef θ sur le k-ième nombres premiers et grandes valeurs de la fonction ω(n) nombre de diviseurs premiers de n
Acta Arith., 42, 367--389.
]:
Theorem (1983)
For any integer n3, the number of prime divisors ω(n) of n satifies:
ω(n)1.3841lognloglogn.
From [Nicolas & Robin, 1983 †Nicolas, J.-L., & Robin, G. 1983
Majorations explicites pour le nombre de diviseurs de n
Canad. Math. Bull., 39, 485--492.
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Theorem (1983)
For any integer n3, the number τ(n) of divisors of n satifies:
τ(n)n1.5379log2/loglogn.
From [Robin, 1983b †Robin, G. 1983b
Méthodes d'optimisation pour un problème de théorie des nombres
RAIRO Inform. Théor., 17, 239--247.
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Theorem (1999)
For any integer n3, we have
τ3(n)n1.59141log3/loglogn
where τ3(n) is the number of triples (d1,d2,d3) such that d1d2d3=n.
From [Duras et al., 1999 †Duras, J.-L., Nicolas, J.-L., & Robin, G. 1999
Grandes valeurs de la fonction {dk}
Pages 743--770 of: Number theory in progress, Vol. 2 (Zakopane-Kościelisko, 1997). Berlin: de Gruyter.
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Theorem (1999)
For any integer n1, any real number s>1 and any integer k1, we have
τk(n)nsζ(s)k1
where τk(n) is the number of k-tuples (d1,d2,,dk) such that d1d2dk=n.
The same paper also announces the bound for n3 and k2
τk(n)naklogk/loglogk
where ak=1.53797logk/log2 but the proof never appeared. From [Nicolas, 2008 †Nicolas, J.-L. 2008
Quelques inégalités effectives entre des fonctions arithmétiques
Functiones et Approximatio, 39, 315--334.
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Theorem (2008)
For any integer n3, we have
σ(n)2.59791nloglog(3τ(n)),
σ(n)n{eγloglog(eτ(n))+logloglog(eeτ(n))+0.9415}.
The first estimate above is a slight improvement of the bound
σ(n)2.59nloglogn(n7)
obtained in [Ivić, 1977 †Ivić, A. 1977
Two inequalities for the sum of the divisors functions
Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak., 7, 17--21.
]. In this same paper, the author proves that
σ(n)2815nloglogn(n31)
where σ(n) is the sum of the unitary divisors of n, i.e. divisors d of n that are such that d and n/d are coprime. ć

Last updated on August 8th, 2012, by Olivier Bordellès