Explicit pointwise upper bounds for some arithmetic functions

The following bounds may be useful is applications.

From [Robin, 1983]:

Theorem (1983)

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

From [Nicolas and Robin, 1983]:

Theorem (1983)

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

From page 51 of [Robin, 1983]:

Theorem (1983)

For any integer \(n\ge3\), we have \(\displaystyle \tau_3(n)\le n^{1.59141 \log 3/\log\log n}\) 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] 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 \(3\le k\le 25\).

Theorem (1999)

For any integer \(n\ge1\), any real number \(s>1\) and any integer \(k\ge1\), we have \(\displaystyle \tau_k(n)\le n^s\zeta(s)^{k-1}\) 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 \(k\ge2\)

\(\displaystyle \tau_k(n)\le n^{a_k\log k/\log\log k}\) where \(a_k=1.53797\log k / \log 2\) but the proof never appeared.

From [Nicolas, 2008]:

Theorem (2008)

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

The first estimate above is a slight improvement of the bound

\(\displaystyle \sigma(n)\le 2.59 n\log\log n\quad(n\ge7)\) obtained in [Ivić, 1977]. In this same paper, the author proves that

\(\displaystyle \sigma^*(n)\le \frac{28}{15} n\log\log n\quad(n\ge31)\) 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 and Koo, 2015] we find the next estimate

Theorem (2015)

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

Further estimates restricted to some sets of integers may be found in this paper as well as in [Washington and Yang, 2021].

On this subject, the readers may consult the web site

The sum of divisors function and the Riemann hypothesis. .