Explicit pointwise upper bounds for some arithmetic functions
=============================================================

.. if-builder:: html

   .. toctree::
      :maxdepth: 2
 
 

The following bounds may be useful is applications.

From
:cite:`Robin83-1`:


.. admonition:: Theorem (1983)
   :class: thm-tme-emt

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


From
:cite:`Nicolas-Robin83`:


.. admonition:: Theorem (1983)
   :class: thm-tme-emt

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


From page 51 of :cite:`Robin83-0`:


.. admonition:: Theorem (1983)
   :class: thm-tme-emt

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


The PhD
memoir
:cite:`Duras93`
contains result concerning the maximum
of :math:`\tau_k(n)`, i.e. the number of :math:`k`-tuples :math:`(d_1,d_2,\ldots, d_k)`
such that :math:`d_1d_2\cdots d_k=n`, when :math:`3\le k\le 25`.


 
  From
  :cite:`Duras-Nicolas-Robin99`:


.. admonition:: Theorem (1999)
   :class: thm-tme-emt

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

The same paper also announces the bound for :math:`n\ge3` and :math:`k\ge2`


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

From :cite:`Nicolas08`:


.. admonition:: Theorem (2008)
   :class: thm-tme-emt

   For any integer :math:`n\ge3`, we have
   :math:`\displaystyle  \sigma(n)\le 2.59791\, n\log\log(3\tau(n)),` 
   :math:`\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
  

:math:`\displaystyle  \sigma(n)\le 2.59 n\log\log n\quad(n\ge7)` 
obtained in
:cite:`Ivic77`.
In this same paper, the author proves that


:math:`\displaystyle  \sigma^*(n)\le \frac{28}{15} n\log\log n\quad(n\ge31)` 
where :math:`\sigma^*(n)` is the sum of the unitary divisors of :math:`n`,
i.e. divisors :math:`d` of :math:`n` that are such that :math:`d` and :math:`n/d` are coprime.


 
In
:cite:`Eum-Koo15`
we find the next estimate

.. admonition:: Theorem (2015)
   :class: thm-tme-emt

   For any integer :math:`n\ge21`, we have
   :math:`\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
:cite:`Washington-Yang21`.


 

 
On this subject, the readers may consult the web site

The sum of divisors function and the
Riemann hypothesis.
.