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Computation of the Mertens constants in arithmetic progressions
A. Languasco
and
A. Zaccagnini
In this page we collect some links concerning the computation of the
Mertens constants in arithmetic progressions.
In a recent paper [1], we proved an elementary formula for the Mertens
constants in arithmetic progressions. These constants are connected with
the asympotic behaviour of the Mertens product in arithmetic progressions.
Such an elementary formula makes it possible to compute the constants using
suitable values of Dirichlet Lseries, see [4].
The actual computations were performed using the following software
on the
NumLab
pcs of the Department of Pure and Applied
Mathematics of the University of Padova.
I have to state the obvious
fact that if you wish to use some of the softwares below for your own research,
you should acknowledge the author and cite the relevant paper in which the program
was used first. In other words, you can use them but you have to
cite the paper of mine that contains such programs.
If you are wondering why I am stating something so trivial, please have a look at P0 here:
A.LanguascoPrograms
Since several people started to use these programs
without citing our paper, I decided to remove them
from this site. (08/19/2022)
The software is now GPLlicensed and with a DOI:
10.24433/CO.5932451.v2
A Code Ocean capsule (able to run an example of use of such a program)
is here:
Software
MertensConstantsfinal.gp:
Pari/GP
script. It can be used via
gp2c.
Input: an integer q, 3≤q≤100.
Output: the Mertens constants C(q,a) (with a precision of at least 100 decimal digits)
for every a≤q such that (a,q)=1.
MCchecks.gp:
Pari/GP
script. It checks that the computed constants
collected in the matrix
MCmatrix.gp
verify eq.(25) of the paper [4] with a precision of at least
100 decimal digits.
MCcheckresults.txt:
text file. Output of MCchecks.gp.
MCfinalresults.pdf:
pdf file containing the results computed using
MertensConstantsfinal.gp.
MCtiming.pdf:
pdf file containing the computation time for the results
computed using MertensConstantsfinal.gp.
References
The papers connected with this computational project are the following ones together
with the references listed there.
[1] A. Languasco, A. Zaccagnini 
A note on Mertens' formula for arithmetic progressions
 Journal of Number Theory,
127
(2007), 3746.
[2]
A. Languasco, A. Zaccagnini 
Some estimates for the average of the error term
of the Mertens product for arithmetic progressions
 Functiones et Approximatio,
Commentarii Mathematici, 38
(2008), 4148.
[3] A. Languasco, A. Zaccagnini 
On the constant in the Mertens product
for arithmetic progressions. I, Identities
 preprint 2007, arxiv:0706.2807.
[4]
A. Languasco, A. Zaccagnini 
On the constant in the Mertens formula for arithmetic progressions. II. Numerical values
 Math. Comp.
78
(2009), 315326.
[5] Pari/GP, version 2.3.1  2.3.2, Bordeaux, 2005,
http://pari.math.ubordeaux.fr/
Other researcher's papers
As I expected, it turned out that these values and/or softwares were useful to other researchers; so far they were used in
the following papers:
K.D. Boklan; J.H. Conway 
Expect at most one billionth of a new Fermat Prime! 
this paper was published on the Math. Intelligencer 39 (2017), pp. 35, but the reference to our
work is just in its arxiv version.
Y. Lamzouri 
A bias in Mertens' product formula 
International Journal of Number Theory, 12
(2016), 97109.
K. Ford; F. Luca; P. Moree 
Values of the Euler phifunction not divisible by a given odd prime,
and the distribution of EulerKronecker constants for cyclotomic fields  Math. Comp. 83 (2014), 14571476.
S.A. Fletcher; P.P. Nielsen; P. Ochem 
Sieve methods for odd perfect numbers 
Math. Comp. 81 (2012), 17531776.
Y. Lamzouri; M.T. Phaovibul; A. Zaharescu 
On the distribution of
the partial sum of Euler's totient function in residue classes 
Colloq. Math. 123 (2011), 115127.
S.R. Finch; P. Sebah 
Residue of a Mod 5 Euler Product 
arxiv, 2009.
S.R. Finch 
Mertens' Formula 
preprint, 2007.
Changes in this page:
02/01/08: correction of some misprints in this page.
20/02/08: correction of some misprints in the message outputs
of MertensConstantsfinal.gp; updated reference [4].
10/03/08: correction of some misprints in the message outputs
of MertensConstantsfinal.gp, MCchecks.gp and in the running titles of
MCtiming.pdf and MCfinalresults.pdf.
16/10/08: updated references [2] and [4].
Aug. 16th 2016: added section about other researcher's papers.
Jan. 26th 2021: since these constants were inserted in
OEIS
and the Pari/GP script mentioned before is cited too, I
checked if it was able to run on a recent version of parti/gp.
Just changing one line to avoid a nameconflict,
such a script runs smoothly on Pari/GP 2.13.1, gp2c 0.0.12,
with an impressive improvement in its running time
(comparing with the ones obtained in 2007).
Clearly now an analogous script can be written using internal
functions of Pari/GP that at that time were not
available (for example: the Dirichlet characters generation
and the Bernoulli polynomials computation.)
Ultimo aggiornamento: 09.12.2022: 13:48:57
