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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 Kummer ratio for prime cyclotomic fields - reprise
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 Kummer ratio for prime cyclotomic fields.
For the mathematical description of the problem and of the quantities R(q) and r(q), please refer to [1].
These computations are part of a joint project with N. Kandhil, S. Saad Eddin, P. Moree and A. Sedunova and it is an updated version of what presented into this old version, (see also [4]).
The difference with the old version is that the computation of the records were performed using the 128 bits precision of the C language (using FFTW).
In particular in [1] we established that
thus getting a new record for both maximal and minimal values of R(q), see [1], Section 6.
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
Kummer-Bernoulli-final.gp: Pari/GP script. It can be used via gp2c. The function to be run is:
global_kummer_bernoulli (r1,r2,defaultprecision).
Input: 2< r1 < r2, two integers; defaultprecision is the number of digits requested.
Output: the triplet: [R(q), r(q) = log R(q), log h1(q)] for every odd prime q such that r1≤q≤r2.
Comments: it computes the sequence gk mod q, where g is a primitive root of Zq*. Then it obtains the generalised Bernoulli numbers and hence R(q) with a trivial implementation of the sum over a, 1≤a≤q-1. Examples on how to use the function and computational results are collected towards the end of the file.
Kummer-Bernoulli-final_DIF.gp: As the previous one, but with the Decimation in Frequency (DIF) strategy implemented.
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 R(q) were computed using the same strategy already used for [2]-[3], 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, it was developed a 128-bits version of these programs which was used for the values described in Table 2 and 3 of [1].
Results
The results presented in [1] can be retrieved as follows.
The ones for q up to 401179 are contained towards the end of the each gp script listed before; if you compare the running times, you'll see that the DIF version is faster (and it also uses less memory).
The others were obtained with the C programs (and the FFTW library - 128 bits precision) are collected in the directory results.
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 R(q) and r(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 - The Kummer ratio of the relative class number for prime cyclotomic fields - J. Math. Anal. Appl. 538(1):Paper No. 128368, 2024.
[2] A. Languasco - Efficient computation of the Euler-Kronecker constants for prime cyclotomic fields - Research in Number Theory 7 (2021), no. 1, Paper no. 2.
[3] 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.
[4] A. Languasco, P. Moree, S. Saad Eddin, A. Sedunova - Computation of the Kummer ratio of the class number for prime cyclotomic fields , Arxiv, 2019.
Ultimo aggiornamento: 28.09.2024: 10:48:12
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