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## Computation of the Mertens and Meissel-Mertens 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(1-1/p)+1/p over primes in arithmetic progressions (B(q,a) which is also called the Meissel-Mertens 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 ``On-Line 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.Languasco-Programs

Software: B, the Meissel-Mertens constant

First we need to compute the Meissel-Mertens constant sum of log(1-1/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 Meissel-Mertens 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
Msum-progressions.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 Msum-progressions.gp.
timeM.pdf:   pdf file containing the computation time for the results computed using Msum-progressions.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
B-qcontrol.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 B-qcontrol.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.
B-direct-timing.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 Msum-progressions.gp can be easily adapted to this goal: the key point is to increase the GP-defaultprecision 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), 37-46.
[2] A. Languasco, A. Zaccagnini - On the constant in the Mertens product for arithmetic progressions. I, Identities Functiones et Approximatio, Commentarii Mathematici, 42 (2010), 17-27.
[3] A. Languasco, A. Zaccagnini - On the constant in the Mertens formula for arithmetic progressions. II. Numerical values  - Math. Comp. 78 (2009), 315-326.
[4] A. Languasco, A. Zaccagnini - Computing the Mertens and Meissel-Mertens constants for sums over arithmetic progressions - with an Appendix by Karl. K. Norton - Experimental Mathematics, 19.3 (2010), 279-284
[5] OEIS: sequence A161529.
[6] PARI/GP, version 2.3.1 - 2.3.5, Bordeaux, 2005, http://pari.math.u-bordeaux.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), 565-575.
X. Meng - Large bias for integers with prime factors in arithmetic progressions  - Mathematika 64 (1) 2018, 237-252.
J. Bellaiche; J.L. Nicolas - Parite des coefficients de formes modulaires  - The Ramanujan Journal, 40 (2016), 1-44.
K. Ford; F. Luca; P. Moree - Values of the Euler phi-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields - Math. Comp. 83 (2014), 1457-1476.
S.A. Fletcher; P.P. Nielsen; P. Ochem - Sieve methods for odd perfect numbers - Math. Comp. 81 (2012), 1753-1776.
R. Baillie - Fun with very large numbers - arxiv, (2011).

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.

Ultimo aggiornamento: 10.12.2021: 15:13:41

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