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Papers:
List Papers;
(with Abstracts);
Curriculum (in Italian):
long version ;
short version;
in English:
long version.
Google Scholar profile.
ResearchGate page.
Orcid ID.
Scopus Author ID.
Web of Science Researcher ID,
Mathematical Reviews page,
Zentralblatt page,
IRIS-CINECA bibliometric parameters (italian ASN) [2023].
English C1 badge.
Computation of Euler constants from primes in arithmetic progressions
A. Languasco
(for a joint paper with P. Moree)
In this page we collect some links concerning the computation of Euler constants in arithmetic progressions.
For the mathematical description of the problem and of the quantities γ(d,a), please refer to [1].
These computations are part of a joint project with Pieter Moree.
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
Pari/GP scripts To compute γ(d,a) we have the following program:
[A]: gamma_arprog.gp: Pari/GP script. It can be used via gp2c. CodeOcean Capsule
There are two functions dedicated to this problem. The first is:
AP_gamma (d,defaultprecision).
Input: 3 ≤ d ≤ 1000, an integer; defaultprecision (≥2) is the number of correct decimals requested.
Output: the values: γ(d,a) for every 1 ≤ a ≤ d-1, (a,d)=1.
The second is: AP_gamma_all (D,defaultprecision).
Input: 3 ≤ D ≤ 1000, an integer; defaultprecision (≥2) is the number of correct decimals requested.
Output: ALL the the values: γ(d,a) for every 1 ≤ a ≤ d-1, (a,d)=1, and every 3≤ d ≤ D.
--------------
To compute the other sums mentioned in Remark 1 of [1] we have the following programs:
[B]: sumprimes_derlog_AP.gp: Pari/GP script. It can be used via gp2c.
It computes Σ p ≡ a mod d p^u log (p) / (p^s-1) for Re(s) > Re(u) + 1.
sum_primes_derlog_AP(d, u, s, prec=19).
Input: 3 ≤ d ≤ 1000, an integer; u,s: complex numbers having Re(s)> Re(u)+1; defaultprecision (≥2) is the number of correct decimals requested.
Output: the values: Σ p ≡ a mod d p^u log (p) / (p^s-1) for every 1 ≤ a ≤ d-1, (a,d)=1.
[C]: lazygamma_AP.gp: Pari/GP script. It can be used via gp2c.
It computes Σ p ≡ a mod d log (p) / (p^s*(p-1)) for Re(s)>0.
lazygamma_AP(d, s, prec=19).
Input: 3 ≤ d ≤ 1000, an integer; s: a complex number having Re(s)>0; defaultprecision (≥2) is the number of correct decimals requested.
Output: the values: Σ p ≡ a mod d log (p) / (p^s*(p-1)) for every 1 ≤ a ≤ d-1, (a,d)=1
[D]: lazygamma.gp: Pari/GP script. It can be used via gp2c.
It computes Σ p log (p) / (p^s*(p-1)) for Re(s)>0.
lazygamma(s, prec=19).
Input: s: a complex number having Re(s)>0; defaultprecision (≥2) is the number of correct decimals requested.
Output: the value: Σ p log (p) / (p^s*(p-1)) for Re(s)>0.
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To compute the Euler products in AP of Section 7 we have the following program:
sumprimes-for-product_AP.gp: Pari/GP script. It can be used via gp2c.
Σ p ≡ a mod d log( 1- 1/p^s) for Re(s) > 1.
SumprimesAP_log(d, s, prec=19).
Input: 3 ≤ d ≤ 1000, an integer; s: complex numbers having Re(s)> 1; defaultprecision (≥2) is the number of correct decimals requested.
Output: the values: Σ p ≡ a mod d log( 1- 1/p^s) for every 1 ≤ a ≤ d-1, (a,d)=1.
Results
The results presented in [1] are collected in the directory results.
References
Some of the papers connected with this computational project are the following.
[1] A. Languasco, P. Moree - Euler constants from primes in arithmetic progression - to appear in Mathematics of Computation, 2025. Code Ocean capsule.
Ultimo aggiornamento: 17.12.2024: 08:14:59
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