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Explicit bounds on primes
=========================

.. if-builder:: html

   .. toctree::
      :maxdepth: 2




Collecting references: :cite:`Dusart98`, :cite:`Dusart16`


1. Bounds on primes, in special ranges
--------------------------------------

The paper
:cite:`Rosser-Schoenfeld62`,
contains several bounds valid only when the variable is small
enough. These are proved by direction verifications. Different direct computations can be used to get effect in limited range. Here is a typical result.

 
.. admonition:: Theorem (:cite:`Buthe16`)
   :class: thm-tme-emt

   Assume the Riemann Hypothesis has been checked up to height
   :math:`H_0`. Then when :math:`x` satisfies :math:`\sqrt{x/\log x}\le H_0/4.92`, we have
   
   * :math:`|\psi(x)-x|\le \frac{\sqrt{x}}{8\pi}\log^2x` when :math:`x > 59`,
   * :math:`|\theta(x)-x|\le \frac{\sqrt{x}}{8\pi}\log^2x` when :math:`x > 599`,
   * :math:`|\pi(x)-\text{li}(x)|\le \frac{\sqrt{x}}{8\pi}\log x` when :math:`x > 2657`.
   
If we use the value :math:`H_0=3 \cdot 10^{12}` obtained by Platt and Trudgian in :cite:`Platt-Trudgian21a`
(which improves previous results in :cite:`Platt17` ), these
bounds are thus valid for :math:`x\le 2.16\cdot 10^{25}`. In :cite:`Buthe18`
the following bounds are also obtained.

 
.. admonition:: Theorem (2018)
   :class: thm-tme-emt

   We have
   
   * :math:`|\psi(x)-x|\le 0.94\sqrt{x}` when :math:`11 < x\le 10^{19}`,
   * :math:`0<\text{li}(x)-\pi(x)\le\frac{\sqrt{x}}{\log x}\Bigl(1.95+\frac{3.9}{\log x}+\frac{19.5}{\log^2x}\Bigr)`  when :math:`2\le x\le 10^{19}`.
   




2. Bounds on primes, without any congruence condition
-----------------------------------------------------


The subject really started with the four papers
:cite:`Rosser41`, :cite:`Rosser-Schoenfeld62`,
:cite:`Rosser-Schoenfeld75` and :cite:`Schoenfeld76`.

We recall the usual notation: :math:`\pi(x)` is the number of primes up to
:math:`x` (so that :math:`\pi(3)=2`), the function :math:`\psi(x)` is the summatory
function of the van Mangold function :math:`\Lambda`,
i.e. :math:`\psi(x)=\sum_{n\le x}\Lambda(n)`, while we also define
:math:`\vartheta(x)=\sum_{p\le x}\log p`.
Here are some elegant bounds that one can find in those papers.

 
.. admonition:: Theorem (1962)
   :class: thm-tme-emt

   
   * For :math:`x > 0`, we have :math:`\psi(x)\le 1.03883 x` and the maximum of :math:`\psi(x)/x` is attained at :math:`x=113`.
   * When :math:`x\ge17`, we have :math:`\pi(x) > x/\log x`.
   * When :math:`x > 1`, we have :math:`\displaystyle \sum_{p\le x}1/p >  \log\log x`.
   * When :math:`x > 1`, we have :math:`\displaystyle \sum_{p\le x}(\log p)/p < \log x`.
   


There are many other results in those papers.
In
:cite:`Dusart99-1` ,
one can find among other things the inequality

.. raw:: html

   <div align="center">
   $$
   \displaystyle  \pi(x) > \frac{x}{\log x -1},  \text{for} \  x\ge17,
   $$
   </div>
 
and also

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

   
   *  When :math:`x\ge e^{22}`, we have :math:`\displaystyle\psi(x)=x+O^*\Bigl(0.006409\frac{x}{\log x}\Bigr)`.
   *  When :math:`x\ge 10\,544\,111`, we have :math:`\displaystyle\vartheta(x)=x+O^*\Bigl(0.006788\frac{x}{\log x}\Bigr)`.
   *  When :math:`x\ge 3\,594\,641`, we have :math:`\displaystyle\vartheta(x)=x+O^*\Bigl(0.2\frac{x}{\log^2   x}\Bigr)`.
   *  When :math:`x > 1`, we have :math:`\displaystyle\vartheta(x)=x+O^*\Bigl(515\frac{x}{\log^3 x}\Bigr)`.
   


This is improved in :cite:`Dusart16` ,
and in particular, it is shown that the 515 above can be replaced by
20.83 and also that

.. raw:: html

   <div align="center">
   $$
   \displaystyle  \vartheta(x)=x+O^*\Bigl(\frac{x}{\log^3 x}\Bigr),  \text{for} \  x\ge89\,967\,803.
   $$
   </div>

Bounds of the shape :math:`|\psi(x)-x|\le \epsilon x` have started appearing
in
:cite:`Rosser-Schoenfeld62`
The latest paper is
:cite:`Kadiri-Faber13`
with its corrigendum
:cite:`Kadiri-Faber18`,
where the explicit density estimate from
:cite:`Kadiri13`
is put to contribution, even for moderate
values of the variable. In particular, when :math:`x\ge 485\,165\,196`, we have 

.. raw:: html

   <div align="center">
   $$
   \displaystyle \psi(x)=x+O^*(0.00053699\,x).
   $$
   </div>

Sharper estimates of this type  for larger :math:`x` were obtained in
:cite:`Fiori-Kadiri-Swidinsky23` and :cite:`Johnston-Yang23`. In particular, we have :cite:`Johnston-Yang23`
 
.. raw:: html

   <div align="center">
   $$
   \displaystyle   \psi(x)=x+O^*(8.14 \cdot 10^{-20}\,x),  \text{for} \  x\ge 5000.
   $$
   </div>


In
:cite:`Platt-Trudgian21b` ,
one find the following estimate

 
.. admonition:: Theorem (2021)
   :class: thm-tme-emt

   When :math:`x\ge e^{2000}`, we have
   :math:`\biggl|\frac{\psi(x)-x}{x}\biggr|\le 235\,(\log x)^{0.52}\exp-\sqrt{\frac{\log x}{5.573412}}\;.`


The paper
:cite:`Johnston-Yang23`
further proves in Theorem 1.1 the next estimate.

 
.. admonition:: Theorem (2023)
   :class: thm-tme-emt

   When :math:`x\ge 2`, we have
   :math:`\biggl|\frac{\psi(x)-x}{x}\biggr|\le 9.39\,(\log x)^{1.51}\exp-0.8274\sqrt{\log x}\;.`
   
   When :math:`x\ge 23`, we have
   :math:`\biggl|\frac{\psi(x)-x}{x}\biggr|\le 0.026\,(\log x)^{1.801}\exp-0.1853\frac{(\log x)^{3/5}}{(\log\log x)^{1/5}}\;.`

Several estimates are also provided for :math:`\psi(x)`, :math:`\pi(x)` and :math:`\vartheta(x)`.   

A lighly sharper estimate than that above was obtained at the same time in :cite:`Fiori-Kadiri-Swidinsky23`

.. admonition:: Theorem (2023)
   :class: thm-tme-emt

   When :math:`x\ge 2`, we have
   :math:`\biggl|\frac{\psi(x)-x}{x}\biggr|\le 9.22022\,(\log x)^{1.5}\exp-0.8476836\sqrt{\log x}\;.`
   


Refined bounds for :math:`\pi(x)` are to be found in
:cite:`Panaitopol00`
and in
:cite:`Axler16`

By comparing in an efficient manner with :math:`\psi(x)-x`,
:cite:`Ramare12-1`,
obtained the next two results. [There was an error in this paper which
is corrected below].


 
.. admonition:: Theorem (2013)
   :class: thm-tme-emt

   For :math:`x > 1`, we have
   :math:`\sum_{n\le x}\Lambda(n)/n=\log x-\gamma+O^*(1.833/\log^2x)`.
   When :math:`x\ge 1\,520\,000`, we can replace the error term by
   :math:`O^*(0.0067/\log x)`. When :math:`x\ge 468\,000`, we can replace the error term by :math:`O^*(0.01/\log x)`. 
   Furthermore, when :math:`1\le x\le 10^{10}`, this error term can be
   replaced by :math:`O^*(1.31/\sqrt{x})`. 




 
.. admonition:: Theorem (2013)
   :class: thm-tme-emt

   For :math:`x\ge 8950`, we have
   :math:`\displaystyle    \sum_{n\le x}\Lambda(n)/n=\log x-\gamma +\frac{\psi(x)-x}{x}+O^*\Bigl(\frac{1}{2\sqrt{x}}+1.75\cdot 10^{-12}\Bigl)` .



:cite:`Vanlalnagaia15-1`
developed the method to incorporate more functions (and corrected the
initial work), extending results of
:cite:`Rosser-Schoenfeld62` . Here are some of his results.


 
.. admonition:: Theorem (2017)
   :class: thm-tme-emt

   For :math:`x\ge 2`, we have
   :math:`\displaystyle   \sum_{p\le x}\frac1p=\log\log  x+B+O^*\Bigl(\frac{4}{\log^3x}\Bigr).` 
   When :math:`x\ge 1000`, one can replace the 4 in the error term by 2.3,
   and when :math:`x\ge24284`, by 1. The constant :math:`B \approx 0.261497` is known as the Meissel-Mertens constant.





 
.. admonition:: Theorem (2017)
   :class: thm-tme-emt

   For :math:`\log x\ge 4635`, we have
   :math:`\sum_{p\le x}\frac1p=\log\log x+B+O^*\Bigl(1.1\frac{\exp(-\sqrt{0.175\log x})}{(\log x)^{3/4}}\Bigr).`








 
When truncating sums over primes, Lemma 3.2 of :cite:`Ramare13-d`    is handy.

 
.. admonition:: Theorem (2016)
   :class: thm-tme-emt

   Let :math:`f` be a :math:`C^1` non-negative, non-increasing function over
   :math:`[P,\infty)`, where :math:`P\ge 3\,600\,000` is a real number and such
   that :math:`\lim_{t\rightarrow\infty}tf(t)=0`. 
   We have

   :math:`\sum_{p\ge P} f(p)\log p\le (1+\epsilon) \int_P^\infty f(t) dt  +  \epsilon P f(P)  +  P f(P) / (5 \log^2 P)`
   
   with :math:`\epsilon=1/914`. When we can only ensure :math:`P\ge2`, then a similar
   inequality holds, simply replacing the last :math:`1/5` by 4.



The above result  relies on (5.1*) of
:cite:`Schoenfeld76`
because it is easily accessible. However on using
Proposition 5.1 of
:cite:`Dusart07`,
one has access to :math:`\epsilon=1/36260`.


 
  
 
Here is a result due to 
:cite:`Trevino12`.

 
.. admonition:: Theorem (2012)
   :class: thm-tme-emt

   For :math:`x` a positive real number. If :math:`x \geq x_0` then there exist :math:`c_1`
   and :math:`c_2` depending on :math:`x_0` such that
   :math:`\displaystyle  \frac{x^2}{2\log{x}} +\frac{c_1 x^2}{\log^2{x}} \leq \sum_{p \leq x} p \leq\frac{x^2}{2\log{x}} + \frac{c_2 x^2}{\log^2{x}}.` 
   The constants
   :math:`c_1` and :math:`c_2` are given for various values of :math:`x_0` in the next
   table.
   
   :math:`\begin{array}{rrr} x_0 & c_1 & c_2 \\315437   & 0.205448  &  0.330479\\468577   & 0.211359  &  0.32593\\486377   & 0.212904  &  0.325537\\644123   & 0.21429   &  0.322609\\678407   & 0.214931  &  0.322326\\758231   & 0.215541  &  0.321504\\758711   & 0.215939  &  0.321489\\10544111 & 0.239818  &  0.29251\\\end{array}`

     


Further estimates can be found in :cite:`Axler19` (Proposition 2.7 and Corollary 2.8). In :cite:`Deleglise-Nicolas19a`
(Proposition 2.7 and Corollary 2.8) and :cite:`Deleglise-Nicolas19b`
(Proposition 2.5, Corollary 2.6, 2.7 and 2.8), we find among other results the next two.
 
 
.. admonition:: Theorem (2019)
   :class: thm-tme-emt

   On setting :math:`\pi_r(x)=\sum_{p\le x}p^r`, we have

   :math:`\displaystyle    \frac{3x^2}{20(\log x)^4}\le \pi_1(x)-\biggl( \frac{x^2}{2\log x} +\frac{x^2}{4(\log x)^2} \frac{x^2}{4(\log x)^3}\biggr)\le  \frac{107x^2}{160(\log x)^4}` 
  
   where the upper estimate is valid when :math:`x\ge 110117910`
   and the lower one when :math:`x\ge905238547`.
   
   We have
   
   :math:`\displaystyle   \frac{-1069x^3}{648(\log x)^4} \le \pi_2(x)-\biggl( \frac{x^3}{3\log x} +\frac{x^3}{9(\log x)^2}  \frac{x^3}{27(\log x)^3} \biggr)\le \frac{11181x^3}{648(\log x)^4}` 
  
   where the upper estimate is valid when :math:`x\ge 60173`
   and the lower one when :math:`x\ge 1091239`. In addition,
   
   :math:`\displaystyle        \pi_3(x)\le 0.271\frac{x^4}{\log x}\quad\text{for }x\ge 664,` 
 
   :math:`\displaystyle        \pi_4(x)\le 0.237\frac{x^5}{\log x}\quad\text{for }x\ge 200,` 
 
   :math:`\displaystyle        \pi_5(x)\le 0.226\frac{x^5}{\log x}\quad\text{for }x\ge 44,` 
 
   For :math:`x\ge 1` and :math:`r\ge5`, we have
   :math:`\displaystyle   \pi_r(x)\le \frac{\log 3}{3}\bigl(1+(2/3)^r\bigr) \frac{x^{r+1}}{\log x}` .






  

3. Bounds containing the :math:`n`-th prime
-------------------------------------------



Denote by :math:`p_n`  the :math:`n`-th prime. Thus :math:`p_1=2,\;p_2=3,\; p_4=5,\cdots`.

The classical form of the prime number theorem yields easily
:math:`p_n \sim n \log n.`
:cite:`Rosser38`
shows that this equivalence does not oscillate
by proving that :math:`p_n` is greater than :math:`n\log n` for :math:`n\geq 2`.  


The asymptotic formula for :math:`p_n` can be developped as shown in :cite:`Cipolla02`


.. raw:: html

   <div align="center">
   $$
   \displaystyle   p_n\sim n\left(\log n+\log\log n -1+\frac{\log\log n-2}{\log n}-\frac{(\ln\ln n)^2-6\log\log n +11}{2\log^2 n}+\cdots\right).
   $$
   </div>

This asymptotic expansion is the inverse of the logarithmic integral
:math:`\mbox{Li}(x)` obtained by series reversion. 

:cite:`Rosser38`
obtained a finite version of the above: for every :math:`n> 1`:

.. raw:: html

   <div align="center">
   $$
   \displaystyle   n (\log n + \log \log n - 10) < p_n < n (\log n + \log\log n +8).
   $$
   </div>

He improves these results in  :cite:`Rosser41`
:  for every :math:`n\geq 55`,

.. raw:: html

   <div align="center">
   $$
   \displaystyle  n (\log n + \log \log n - 4) < p_n < n (\log n + \log\log n +2).
   $$
   </div>

This result was subsequently improved by Rosser and Schoenfeld :cite:`Rosser-Schoenfeld62` in 1962 to

.. raw:: html

   <div align="center">
   $$
   \displaystyle  n (\log n + \log \log n - 3/2) < p_n < n (\log n + \log\log n -1/2),
   $$
   </div>




where the first and the second inequalities holds for :math:`n > 1` and :math:`n > 19`,
respectively. Those constants  were subsequently reduced by :cite:`Robin83-1` .
In particular, the lower bound 

.. raw:: html

   <div align="center">
    $$
    \displaystyle  n (\log n + \log \log n - 1.0072629) < p_n
    $$
   </div>

is valid for :math:`n>1` and the constant :math:`1.0072629` can be replaced by 1 for
:math:`p_k\leq 10^{11}`.
Then :cite:`Massias-Robin96`
showed that the constant term equals to 1 in the lower bound was admissible for
:math:`p_k < e^{598}`
and :math:`p_k > e^{1800}`. Finally,  :cite:`Dusart99-1`
showed that

.. raw:: html

   <div align="center">
   $$
   \displaystyle    n(\log n - \log \log n - 1) <  p_n,
   $$
   </div>


for all :math:`n>1` and also that

.. raw:: html

   <div align="center">
    $$
    p_n <  n(\log n - \log \log n - 0.9484),
    $$
    </div>


for
:math:`\displaystyle  n > 39017` i.e. :math:`p_n > 467473`.

In
:cite:`Carneiro-Milinovich-Soundararajan19`,
the authors  prove the next result.

.. admonition:: Theorem (2019)
   :class: thm-tme-emt

   Under the Riemann Hypothesis, for all :math:`n \geq 3`,we have, :math:`p_{n+1}-p_n\le\frac{22}{25}\sqrt{p_n}\log p_n`.



 

 
In
:cite:`Axler19`
we find (Theorem 1.6 and 1.7) the next result.

 

.. admonition:: Theorem (2019)
   :class: thm-tme-emt

   We have
   :math:`\displaystyle   \sum_{i\le k}p_i\ge \frac{k^2}4 +\frac{k^2}{4\log k}  -\frac{k^2(\log\log k-2.9)}{4(\log k)^2}\quad(\text{for } k\ge 6309751),` 
   
   as well as
   
   :math:`\displaystyle    \sum_{i\le k}p_i\le\frac{k^2}4 +\frac{k^2}{4\log k} -\frac{k^2(\log\log k-4.42)}{4(\log k)^2}\quad(\text{for }k\ge 256376).` 
  
 
In 
:cite:`DeKoninck-Letendre20`
we find in passing (Lemma 4.8) the next result.

 

.. admonition:: Theorem (2020)
   :class: thm-tme-emt

   We have
   :math:`\displaystyle   \sum_{i\le k}\log p_i\le k\bigl(\log k+\log\log k -1/2\Bigr)\quad(\text{for }k\ge 5),` 
   as well as
   :math:`\displaystyle    \sum_{i\le k}\log\log p_i\ge k\biggl(\log\log k+\frac{\log\log -3/2}{\log k}\biggr)  \quad(\text{for }k\ge 6).` 









4. Bounds on primes in arithmetic progressions
----------------------------------------------


Collecting references:
:cite:`McCurley84-1`, :cite:`McCurley84-2`, :cite:`Ramare-Rumely96`, :cite:`Dusart01`

Lemma 10 of  :cite:`Moree04` ,
section 4 of :cite:`Moree-teRiele04` .

A consequence of Theorem 1.1 and 1.2 of :cite:`Bennett-Martin-OBryant-Rechnitzer18`
states that

 
.. admonition:: Theorem (2018)
   :class: thm-tme-emt

   We have
   
   :math:`\displaystyle \max_{3\le q\le 10^4}\max_{ x\ge 8\cdot 10^9}\max_{\substack{1\le a\le q,\\(a,q)=1}}  \frac{\log x}{x}\Bigl|\sum_{\substack{n\le x,\\ n\equiv a[q]}}\Lambda(n) -\frac{x}{\varphi(q)}\Bigr|\le 1/840.`
   
   When :math:`q\in(10^4, 10^5]`, we may replace :math:`1/840` by :math:`1/160` and when
   :math:`q\ge 10^5`, we may replace :math:`1/840` by :math:`\exp(0.03\sqrt{q}\log^3q)`.
   
   Furthermore, we may replace
   :math:`\sum_{\substack{n\le x,\\ n\equiv a[q]}}\Lambda(n)` by
   :math:`\sum_{\substack{p\le x,\\ p\equiv a[q]}}\log p` with no changes in
   the constants. 


Similarly, as a consequence of Theorem 1.3 of :cite:`Bennett-Martin-OBryant-Rechnitzer18`
states that

 
.. admonition:: Theorem (2018)
   :class: thm-tme-emt

   We have

   :math:`\displaystyle \max_{3\le q\le 10^4}\max_{ x\ge 8\cdot 10^9}\max_{\substack{1\le a\le q,\\(a,q)=1}}  \frac{\log^2 x}{x}\Bigl| \sum_{\substack{p\le x,\\ p\equiv a[q]}}1  -\frac{\textrm{Li}(x)}{\varphi(q)}\Bigr|\le 1/840.`
   
   When :math:`q\in(10^4, 10^5]`, we may replace :math:`1/840` by :math:`1/160` and when
   :math:`q\ge 10^5`, we may replace :math:`1/840` by :math:`\exp(0.03\sqrt{q}\log^3q)`.







 
Concerning estimates with a logarithmic density, in
:cite:`Ramare02-a`
and in
:cite:`Ramare12-0`
estimates for the functions
:math:`\displaystyle\sum_{\substack{n\le x,\\ n\equiv a[q]}}\Lambda(n)/n`
are considered.
Extending computations from the former, the latter paper Theorem 8.1
reads as follows.

 
.. admonition:: Theorem (2016)
   :class: thm-tme-emt

   We have
   :math:`\displaystyle  \max_{q\le 1000}\max_{q\le x\le 10^5}\max_{\substack{1\le a\le q,\\ (a,q)=1}}  \sqrt{x}\Bigl| \sum_{\substack{n\le x,\\ n\equiv a[q]}}\frac{\Lambda(n)}{n}  -\frac{\log x}{\varphi(q)}-C(q,a)\Bigr|\in(0.8533,0.8534)`
   and the maximum is attained with :math:`q=17`, :math:`x=606` and :math:`a=2`.


The constant :math:`C(q,a)` is the one expected, i.e. such that
:math:`\sum_{\substack{n\le x,\\ n\equiv a[q]}}\frac{\Lambda(n)}{n}-\frac{\log x}{\varphi(q)}-C(q,a)` goes to
zero when :math:`x` goes to infinity.

When :math:`q` belongs to "Rumely's list", i.e. in one of the
following set:

    *  :math:`\{k\le 72\}`
    *  :math:`\{k\le 112, k\text{ non premier}\}`
    *  :math:`\begin{aligned}\{116, 117, &120, 121, 124, 125, 128, 132, 140,   143, 144, 156, 163, \\ &169, 180, 216, 243, 256, 360, 420, 432\}\end{aligned}`

Theorem 2 of
:cite:`Ramare02-a`
gives the following.

 
.. admonition:: Theorem (2002)
   :class: thm-tme-emt

   When :math:`q` belongs to Rumely's list and :math:`a` is prime to :math:`q`, we have
   :math:`\displaystyle \sum_{\substack{n\le x,\\ n\equiv a[q]}}\frac{\Lambda(n)}{n}  =\frac{\log x}{\varphi(q)}+C(q,a)+O^*(1)`
   as soon as :math:`x\ge1`.


More precise bounds are given.




5. Least prime verifying a condition
------------------------------------




In :cite:`Bach-Sorenson96` , the authors estimate the magnitude of the first prime number in reduced residue classes.

.. admonition:: Theorem (1996)
   :class: thm-tme-emt

   Let :math:`m` and :math:`q` be integers, with :math:`gcd(m;q)=1`. There is a prime :math:`p \equiv m (\mod q)` satisfying :math:`p < 2(q\log q)^2`.

We also have the following result connected to prime gaps in residue classes :cite:`Kadiri05-2`.

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

   Let :math:`q\geq 3` be a non-exceptional modulus and let :math:`(a,q)=1`. For any
   :math:`\epsilon >0`, there exists a computable constant :math:`\alpha>0` such that, 
   if :math:`x \geq \alpha \log^2q`, then the interval :math:`[e^x,e^{x+\epsilon}]`
   contains a prime :math:`p \equiv a (\mod q)`.

The constant :math:`\alpha` in the theorem above was computed for several :math:`\epsilon` and :math:`q\geq q_0` as in the following table

.. raw:: html

   <div align="center">
   $$ \begin{array}{cccccc} q_0 \backslash \epsilon & 0.0001 & 0.001 & 0.01 & 0.1 & 1 & 10 \\
   5.10^4 & 9.228 & 15.550  &12.245 & 9.4357 & 6.9684 & 4.8430 \\
   10^{10} & 9.8356 & 8.5912 &7.4255 &6.3398 &5.3418 &4.4761\\
   10^{15} &7.6121 &6.8799 &6.1816 &5.5174 &4.8905 &4.3256\\
   10^{20} &6.5919 &6.0799 &5.5864 &5.1114 &4.6565 &4.2373\\
   10^{25} &6.0079 &5.6164 &5.2364 &4.8678 &.5116 &4.1783\\
   10^{30} &5.6298 &5.3137 &5.0053 &4.7047 &4.4123 &4.1357\\
   10^{35} &5.3649 &5.0102 &4.8411 &4.5875 &4.3396 &4.1032\\
   10^{40} &5.1688 &4.9414 &4.7181 &4.4989 &4.2839 &4.0776\\
   10^{45} &5.0178 &4.8185 &4.6225 &4.4295 &4.2398 &4.0567\\
   10^{50} &4.8979 &4.7205 &4.5459 &4.3737 &4.2039 &4.0394\\
   10^{55} &4.8003 &4.6407 &4.4832 &4.3276 &4.1740 &4.0247\\
   10^{60} &4.7192 &4.5742 &4.4308 &4.2890 &4.1488 &4.0121\\
   10^{65} &4.6509 &4.5179 &4.3864 &4.2562 &4.1272 &4.0011\\
   10^{70} &4.5924 &4.4697 &4.3482 &4.2278 &4.1084 &3.9915\\
   10^{75} &4.5418 &4.4280 &4.3151 &4.2031 &4.0920 &3.9829\\
   10^{80} &4.4976 &4.3914 &4.2860 &4.1814 &4.0774 &3.9753\\10^{85} &4.4587 &4.3591 &4.2603 &4.1621 &4.0645 &3.9684\\10^{90} &4.4240 &4.3304 &4.2373 &4.1448 &4.0528 &3.9622\\10^{95} &4.3931 &4.3046 &4.2168 &4.1293 &4.0423 &3.9565\\10^{100} &4.3652& 4.2815 &4.1982 &4.1153 &4.0328 &3.9513 \end{array} $$
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