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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 L-series, 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.


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), 37--46.
[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), 41--48.
[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), 315-326.
[5] PARI/GP, version 2.3.1 - 2.3.2, 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:
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. 3--5, 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), 97-109.
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.
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), 115-127.
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.



Ultimo aggiornamento: 01.04.2017: 06:13:27

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