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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 Zcyc (s) = ∏qζKq(s)
function
and related quantities for the paper
``Coprime and squarefree ideals in infinite families of number fields''
(jointly with Steve Fan, Rashi Lunia and 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 Zeta-function Zcyc(s), s ≥ 2, obtained by multiplying together the Dedekind zeta function ζKq(s), s ≥ 2, where q runs over the prime numbers and Kq= ℚ(ζq) denotes the q-th cyclotomic field and ζq is a primitive q-root of unity.
Each ζKq(s)-value is obtained using its representation as a product of the Riemann zeta-function and of the Dirichlet L-functions attached to the non-principal characters χ defined mod q.
Each Dirichlet L-function values L(s,χ) is then obtained by writing L(s,χ) as a sum over a, a=1,...,q-1, of χ(a)ζ(s,a/q), where ζ(s, x), Re(s)>1, x > 0, denotes the Hurwitz zeta-function.
In a similar way, one can obtains the values of the Z+cyc(s), defined using only the Dedekind zeta function ζK+q(s), where K+q denotes the maximal real subfield of Kq.
Using the values of the first partial derivative of ζ(s,a/q), one can also obtain the values of L'(s,χ), L'(s,χ)/L(s,χ) and hence the ones for ζ'Kq(s), ζ'Kq(s)/ζKq(s), Z'cyc(s)/Zcyc(s) and the analogous ones for the similar objects attached to the maximal real subfield K+q.
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
Zeta_cycl_interval.gp: Pari/GP script. It can be used via gp2c. The function to be called is:
zeta_cyc(mins = 2, maxs = 6, gaps = 0.1, Q=1000, prec=19)
that computes the values of Zcyc(s), Z+cyc(s), Z'cyc(s)/Zcyc(s), Z+'cyc(s)/Z+cyc(s) for s from mins to maxs, into the points mins+k*gaps, using the primes q up to Q.
The internal precision of the computation is fixed to prec decimal digits.
The results are saved into a .csv file.
It was used to compute the data presented in Table 2.
Zcyc_direct.gp: Pari/GP script. It can be used via gp2c. The function to be called is:
Zcyc_direct(P1, P2, prec,start)
that approximates the values of Zcyc(2) using primes from P1 to P2, starting the product with the value start and with an internal precision of the computation fixed to prec decimal digits.
It uses the Pari/gp internal lfun function to define the Dedekind zeta function of Kq= ℚ(ζq).
It was used to compute the data presented in Table 1.
Results
The results presented in [1] are collected in the directory results.
The plots presented in [1] are collected in the directory plots.
References
Some of the papers connected with this project are the following.
[1] Steve Fan, Alessandro Languasco, Rashi Lunia, Pieter Moree, Coprimality in infinite families of number fields - in preparation, 2025.
Ultimo aggiornamento: 06.05.2025: 10:01:45
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