|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 the difference between Euler-Kronecker
constants
for prime cyclotomic fields and their maximal real subfields
A. Languasco
(for a joint paper with N. Kandhil, S. Saad Eddin, P. Moree
and A. Sedunova)
In this page we collect some links concerning the computation of the Euler-Kronecker constants for prime cyclotomic fields and their maximal real subfields.
For the mathematical description of the problem and of the quantities k(q) and r(q), please refer to [1] and [2].
Here you can find the description of the Pari/GP scripts and the C programs used.
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
kappaq-direct-v1.gp: Pari/GP script. It can be used via gp2c. The function to be run is:
global_eulerkronecker_direct (r1,r2,defaultprecision).
Input: 2< r1 < r2, two integers; defaultprecision is the number of digits requested.
Output: the triplet: [γq, γq+, k(q) = γq+-γq] for every odd prime q such that r1≤q≤r2.
Comments: it computes γq, γq+, k(q) using the Pari/GP lfun function and a direct sum over χ. Examples on how to use the function and computational results are collected towards the end of the file. In particular, the values presented in Table 2 of [1] were obtained using this program.
C programs
Examples on how to use the following programs and the results obtained with them are contained in the directory: results.
FFT programs : The values of γq, γq+, k(q) were computed using the same strategy already used for [3]-[4], see- Computation of Ramanujan-Deninger Gamma function and some number theoretic applications- Computation of the Euler-Kronecker constants for prime cyclotomic fields (and the generalised Euler constants in arithmetic progressions)In particular, here we just focused on the interpretation of the quantities γq+-γq and its analogies with r(q).
Python programs
The python programs are used to produce plots and histograms starting from the computed data (contained in the folder python/data).
The plots and the histograms are saved into the folder python/plots_results.
They are python/pgm-rq-analysis.py and python/pgm-kq-analysis.py
They both need python v.3.10 and the packages matplotlib, pandas, numpy, scipy, sympy, humanfriendly.
Results
The results presented in [1] can be retrieved as follows.
The ones for every prime between 3 and 107 (C-FFTW program - 80 bits precision) can be found in a csv file here results; the analysis on this file were performed using a python3-pandas script.
In the directory plots you can find the histograms and the scatter plots of the normalised results for γq, γq+, k(q) for every prime between 3 and 107.
References
Some of the papers connected with this computational project are the following.
[1] N. Khandil, A. Languasco, P. Moree, S. Saad Eddin, A. Sedunova - Relative class numbers and Euler-Kronecker constants of maximal real cyclotomic subfields - arxiv preprint, 2024.
[2] N. Khandil, A. Languasco, P. Moree, S. Saad Eddin, A. Sedunova - The Kummer ratio of the relative class number for prime cyclotomic fields - J. Math. Anal. Appl. 538(1):Paper No. 128368, 2024.
[3] A. Languasco - Efficient computation of the Euler-Kronecker constants for prime cyclotomic fields - Research in Number Theory 7 (2021), no. 1, Paper no. 2.
[4] A. Languasco, L. Righi - A fast algorithm to compute the Ramanujan-Deninger gamma function and some number-theoretic applications - Mathematics of Computation 90 (2021), 2899--2921.
Ultimo aggiornamento: 28.09.2024: 10:46:49
Go to the
DEI Department
Home Page
|
|
|
|
|
|
|
|
|