|
|
|
|
|
|
|
|
|
|
|
|
Computation of the Mertens constants in arithmetic progressions
A. Languasco
and
A. Zaccagnini
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/
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]).
Ultimo aggiornamento: 10.08.2009: 09:03:02
Go to the
Maths Department
Home Page
|
|
|
|
|
|
|