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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,
Arxiv preprints,
IRIS-CINECA bibliometric parameters (italian ASN) [2024].
English C1 badge.
Computation of the relevant quantities presented in the paper
``Quadratic residue bias of the divisor function, Fekete polynomials and prime gaps''
(with Pieter Moree)
In this page I include my programs (Pari/GP scripts) developed to obtain the numerical results described in the paper [1].
The goal is to show the how to compute the quantities ρ(q) and ρ(q)2 -nq for prime numbers q congruent to ± 1 mod 8.
The computations involve a series of Euler products. To evaluate them we heavily use the prodeulerrat function of Pari/GP.
The plots are obtained from the saved data by using the matplotlib package of python.
For more details, please refer to [1].
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
Adivisors-file-v6.gp: Pari/GP script. It can be used via gp2c. The function to be called is:
Aqdivisors(q, defaultprecision)
that computes the values of nq, A(q,1), ρ=(1+A(q,1))/(1-A(q,1)) rho_tilde = ρ(q)2 -nq, rhonalpha , rhonbeta.
The internal precision of the computation is fixed to defaultprecision decimal digits.
The results are saved into a .csv file.
It was used to compute the data presented in Tables 1 and 2 and Figure 1.
nuovipols-v2.gp: Pari/GP script. It can be used via gp2c. The functions to be called are:
firstpols() and severalpols().
The first computes the sandwich bounds presented in the first part of Section 5.1 of [1], the second the ones presented in Table 3 and printed also in Figure 2.
Results
The results presented in [1] are collected in the directory results.
The plots presented in [1] are collected in the directory plots.
The data for Table 3 and Figure 2 presented in [1] are collected in the directory Table 3.
The data for Table 4 presented in [1] are collected in the directory Table 4.
The data for Tables 5-6 presented in [1] are collected in the directory prime_gaps.
The computations used in Proposition 3 in [1] are collected in the directory Proposition 3.
References
Some of the papers connected with this project are the following.
[1] Alessandro Languasco, Pieter Moree, Quadratic residue bias of the divisor function, Fekete polynomials and prime gaps - submitted, 2025.
Ultimo aggiornamento: 09.08.2025: 14:34:56
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