Explicit pointwise upper bounds for some arithmetic functions
The following bounds may be useful is applications. From :
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 :
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 :
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 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 :
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 :
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 . 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 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 .

On this subject, the readers may consult the web site The sum of divisors function and the Riemann hypothesis. .
Last updated on September 19th, 2021, by Olivier Ramaré