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Explicit pointwise upper bounds for some arithmetic functions

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The following bounds may be useful is applications. From [Robin, 1983a †Robin, G. 1983a
Estimation de la fonction de Tchebychef $\theta$ sur le $k$-ième nombres premiers et grandes valeurs de la fonction $\omega(n)$ nombre de diviseurs premiers de $n$
Acta Arith., 42, 367--389.]:
Theorem (1983)

For any integer $n\ge3$, the number of prime divisors $\omega(n)$ of $n$ satisfies: $$\omega(n)\le 1.3841\frac{\log n}{\log\log n}.$$

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.]:
Theorem (1983)

For any integer $n\ge3$, the number $\tau(n)$ of divisors of $n$ satisfies: $$\tau(n)\le n^{1.538 \log 2/\log\log n}.$$

From page 51 of [Robin, 1983b †Robin, G. 1983b
Grandes valeurs de fonctions arithmétiques et problèmes d'optimisation en nombres entiers
Ph.D. thesis, Université de Limoges.]:
Theorem (1983)

For any integer $n\ge3$, we have $$\tau_3(n)\le n^{1.59141 \log 3/\log\log n}$$ where $\tau_3(n)$ is the number of triples $(d_1,d_2,d_3)$ such that $d_1d_2d_3=n$.

The PhD memoir [Duras, 1993 †Duras, J.-L. 1993
Etude de la fonction nombre de façons de représenter un entier comme produit de k facteurs
Ph.D. thesis, Université de Limoges. .] contains result concerning the maximum of $\tau_k(n)$, i.e. the number of $k$-tuples $(d_1,d_2,\ldots, d_k)$ such that $d_1d_2\cdots d_k=n$, when $3\le k\le 25$.
From [Duras et al., 1999 †Duras, J.-L., Nicolas, J.-L., & Robin, G. 1999
Grandes valeurs de la fonction {$d_k$}
Pages 743--770 of: Number theory in progress, Vol. 2 (Zakopane-Kościelisko, 1997). Berlin: de Gruyter.]:
Theorem (1999)

For any integer $n\ge1$, any real number $s>1$ and any integer $k\ge1$, we have $$\tau_k(n)\le n^s\zeta(s)^{k-1}$$ where $\tau_k(n)$ is the number of $k$-tuples $(d_1,d_2,\cdots,d_k)$ such that $d_1d_2\cdots d_k=n$.

The same paper also announces the bound for $n\ge3$ and $k\ge2$ $$ \tau_k(n)\le n^{a_k\log k/\log\log k} $$ where $a_k=1.53797\log k / \log 2$ 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.]:
Theorem (2008)

For any integer $n\ge3$, we have $$\sigma(n)\le 2.59791\, n\log\log(3\tau(n)),$$ $$\sigma(n)\le n\{ e^\gamma\log\log(e\tau(n))+\log\log\log(e^e\tau(n))+0.9415\}.$$

The first estimate above is a slight improvement of the bound $$\sigma(n)\le 2.59 n\log\log n\quad(n\ge7)$$ 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 $$\sigma^*(n)\le \frac{28}{15} n\log\log n\quad(n\ge31)$$ where $\sigma^*(n)$ 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.
In [Eum & Koo, 2015 †Eum, Ick Sun, & Koo, Ja Kyung. 2015
The Riemann hypothesis and an upper bound of the divisor function for odd integers
J. Math. Anal. Appl., 421(1), 917--924.] we find the next estimate
Theorem (2015)

For any integer $n\ge21$, we have $$\sigma(n)\le \tfrac34e^\gamma n\log\log n.$$

Further estimates restricted to some sets of integers may be found in this paper as well as in [Washington & Yang, 2021 †Washington, Lawrence C., & Yang, Ambrose. 2021
Analogues of the Robin-Lagarias criteria for the Riemann hypothesis
Int. J. Number Theory, 17(4), 843--870.].

On this subject, the readers may consult the web site The sum of divisors function and the Riemann hypothesis. .

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