
Papers:
List Papers;
(with Abstracts);
Curriculum (in Italian):
long version ;
short version);
Google Scholar profile.
ResearchGate page.
Orcid ID.
Scopus Author ID.
Thomson Reuters Researcher ID,
Mathematical Reviews page.
Computation of the Mertens and MeisselMertens
constants
for sums over arithmetic progressions
A. Languasco
and
A. Zaccagnini
In this page we collect some links concerning the computation of the
Mertens constants for sum of primes in arithmetic progressions.
In a recent paper [4], we described how to connect three
Mertens constants in arithmetic progressions. These constants are involved in
the asympotic behaviour of the Mertens product in arithmetic progressions
(C(q,a)), in the sum of 1/p over primes in arithmetic progressions (M(q,a))
and in the sum of log(11/p)+1/p over primes in arithmetic progressions
(B(q,a) which is also called the MeisselMertens constant).
It turned out that, for every integer q>=3 and (q,a)=1, M(q,a)
is the easiest computable constant and such a computation, together with
the one we did for C(q,a), leads immediately to get B(q,a).
The sequence M(3,1) was recently inserted in the
``OnLine Encyclopedia of Integer Sequences''
with the id number A161529; see the page:
A161529.
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
Software:
B, the MeisselMertens constant
First we need to compute the MeisselMertens constant
sum of log(11/p)+1/p over primes to be able to
perform the needed verifications on M(q,a) and B(q,a).
MeisselMertens.gp:
PARI/GP
script. It can be used via
gp2c.
Input: the integers A,K,prec: A is the cutting parameter for the finite sum;
K is the cutting parameter for the sum over log(zeta(k)); prec the precision
used by PARI to do the computations.
Output: the MeisselMertens constants B.
Comment: to get about 100 correct digits: set A=100, K=55, prec=120;
to get 1000 correct digits: set A=145, K=465, prec=1030.
Software:
M(q,a), 3<=q<=100, with 100 correct digits
Msumprogressions.gp:
PARI/GP
script. It can be used via
gp2c.
Input: an integer q, 3≤q≤100.
Output: the Mertens constants M(q,a) (with a precision of at least 100 decimal digits)
for every a≤q such that (q,a)=1.
Msumchecks.gp:
PARI/GP
script. It checks that the computed constants
collected in the matrix
Msummatrix.gp
verify a consistency condition with a precision of at least
100 decimal digits.
Msumcheckresults.txt:
text file. Output of Msumchecks.gp.
Msumfinalresults.pdf:
pdf file containing the results computed using
Msumprogressions.gp.
timeM.pdf:
pdf file containing the computation time for the results
computed using Msumprogressions.gp.
Software: recomputation of some values of
C(q,a), 3<=q<=100, with 100 correct digits
We recomputed the cases
q= 17,27,29,34,41,43,46,47,51,53,54,61,65,67,71,72,81,82,86,92,94
to remove an "error propagation" phenomenon in computing B(q,a)
via C(q,a) and M(q,a).
We recomputed the cases
q=59,73,79,83,87,89,93,97
to have the PARI/Gp REALPRECISION parameter =120 for the complete
set of C(q,a) values.
MertensConstantfinalRIC.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 (q,a)=1.
It is the same program used
here
in which we improved the choices of the parameters for 61≤q≤100
to have a faster execution (about twice faster).
Msumchecks.gp:
PARI/GP
script. It checks that the computed constants
collected in the matrix
MCmatrix.gp
verify a consistency condition with a precision of at least
100 decimal digits.
It is the same program used
here.
MCcheckresults.txt:
text file. Output of MCchecks.gp.
Software:
B(q,a), 3<=q<=100, with 100 correct digits (via C(q,a) and M(q,a))
Bqa.gp:
PARI/GP
script. It can be used via
gp2c.
Input: an integer q, 3≤q≤100.
Output: the Mertens constants B(q,a) (with a precision of at least 100 decimal digits)
for every a≤q such that (q,a)=1 computed using M(q,a) and C(q,a) collected
in the matrices
Msummatrix.gp
MCmatrix.gp
Bqcontrol.gp,
Bsumchecks.gp:
PARI/GP
scripts. It checks that the computed constants
collected in the matrix
MBmatrix.gp
verify two consistency conditions with a precision of at least
100 decimal digits.
Bqcheckresults.txt,
Bsumcheckresults.txt:
text files. Outputs of Bqcontrol.gp and Bsumchecks.gp.
Software:
B(q,a), 3<=q<=100, with 100 correct digits (direct computation) and comparisons with the indirect case
Bprogressions.gp:
PARI/GP
script. It can be used via
gp2c.
Input: an integer q, 3≤q≤100.
Output: the Mertens constants B(q,a) (with a precision of at least 100 decimal digits)
for every a≤q such that (q,a)=1.
Bdirecttiming.pdf:
pdf file containing the computation time for the results
computed using Bprogressions.gp.
controlB.gp:
PARI/GP
script. It checks that the constants B(q,a)
computed via M(q,a) and C(q,a) (collected
collected in the matrix
MBmatrix.gp
)
and the ones computed direcly,
Bmatrixdirect.gp
,
are the same with a precision of at least
100 decimal digits.
compareB.txt:
text file. Output of controlB.gp.
Software:
B(q,a), C(q,a) and M(q,a), 3<=q<=300, with 20 correct digits
BCM20.gp:
PARI/GP
script. It can be used via
gp2c.
Input: an integer q, 3≤q≤300.
Output: three files
matricesB.txt:
matricesC.txt:
matricesM.txt:
containing the Mertens constants B(q,a), C(q,a) and M(q,a)
(with a precision of at least 20 decimal digits)
for every a≤q such that (q,a)=1.
Bsumchecks20.gp,
Cchecks20.gp,
Msumchecks20.gp:
PARI/GP
scripts. They check that the computed constants
collected in the matrices
Bmatverif.gp,
Cmatverif.gp,
Mmatverif.gp
verify a consistency condition with a precision of at least
20 decimal digits.
Bverif20.txt,
Cverif20.txt,
Mverif20.txt:
text files. Outputs of Bsumchecks20.gp, Cchecks20.gp, Msumchecks20.gp.
In May 2011,
Robert Baillie asked me for a more precise
evalutation of M(100,1) for his work
``Fun with very large numbers''
. More precisely
he needed more than 110 correct decimal digits. My program
Msumprogressions.gp can be easily adapted to this goal: the key point is to increase
the GPdefaultprecision and to increase the parameters
N,T, K (see [4] for their meaning). This way I was able compute M(100,a), 1<=a<=100,
(a,100)=1,
with more than 130 correct decimal digits.
The results of this computation in the case q=100 and
with 134 correct decimal digits is here
Mertenssum130.txt
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 
On the constant in the Mertens product
for arithmetic progressions. I, Identities
Functiones et Approximatio, Commentarii Mathematici, 42 (2010), 1727.
[3]
A. Languasco, A. Zaccagnini 
On the constant in the Mertens formula for arithmetic progressions. II. Numerical values
 Math. Comp.
78
(2009), 315326.
[4]
A. Languasco, A. Zaccagnini 
Computing the Mertens and MeisselMertens constants for sums over arithmetic progressions
 with an Appendix by Karl. K. Norton  Experimental Mathematics,
19.3 (2010), 279284
[5]
OEIS: sequence A161529.
[6] PARI/GP, version 2.3.1  2.3.5, 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:
T. Yamada 
Quasiperfect numbers with the same exponent
 arxiv, (2016).
C. Elsholtz; S. Planitzer 
On Erdos and Sarkozy's sequences with Property P
 Monatsh. Math. 182,(2017), 565575.
X. Meng 
Large bias for integers with prime factors in arithmetic progressions
 Mathematika 64 (1) 2018, 237252.
J. Bellaiche; J.L. Nicolas 
Parite des coefficients de formes modulaires
 The Ramanujan Journal,
40 (2016), 144.
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.
R. Baillie 
Fun with very large numbers 
arxiv, (2011).
Changes in this page:
Aug. 22nd 2009: add reference to the OEIS.
Apr. 18th 2011: updated reference [4].
May. 6th 2011: added section concerning M(100,a), 1<=a<=100,
(a,100)=1, with more than 130 correct decimal digits for Baillie's paper.
Aug. 16th 2016: added section about other researcher's papers.
Ultimo aggiornamento: 10.12.2021: 15:13:41
