Papers: List Papers; (with Abstracts); Curriculum (in Italian): long version ; short version); Google Scholar profile. ResearchGate page. Orcid ID. Scopus Author ID. Thomson Reuters Researcher ID, Mathematical Reviews page.

Computation of Kummer ratio for prime cyclotomic fields A. Languasco (for a joint paper with Pieter Moree, Sumaia Saad Eddin and Alisa Sedunova)

In this page we collect some links concerning the computation of the Kummer ratio for prime cyclotomic fields. These computations are part of a joint project with Sumaia Saad Eddin, Pieter Moree and Alisa Sedunova. In a recent paper , I (A. Languasco) compared three different methods to compute the Kummer ratio r(q) for prime cyclotomic fields.

In particular in  I established that
r(6766811) = 1.709379042...    and    r(116827429) = 0.575674...

thus getting a new record for both maximal and minimal values of r(q). The previous records were r(5231) = 1.556562... (Shokrollahi ) and r(3) =0.604599...

Here you can find the description of the PARI/Gp scripts and the C programs used. More detailed instructions on how to use the C programs are here: How-it-works.txt.

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-direct-final.gp: PARI/GP script. It can be used via gp2c. The function to be run is:
global_kummer_direct (r1,r2,defaultprecision).
Input: 2< r1 < r2, two integers; defaultprecision is the number of digits requested.
Output: r(q) for every odd prime q such that r1≤q≤r2.
Comment: it uses the lfun command of PARI/Gp and the Conrey description of Dirichlet characters. Examples on how to use the function and computational results are collected towards the end of the file.
Kummer-psi-final.gp: PARI/GP script. It can be used via gp2c. The function to be run is:
global_kummer_psi (r1,r2,defaultprecision).
Input: 2< r1 < r2, two integers; defaultprecision is the number of digits requested.
Output: r(q) for every odd prime q such that r1≤q≤r2.
Comments: it computes first the needed values of ψ at the a/q points, see , and then obtain 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.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: r(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.

C programs
Examples on how to use the following programs and the results obtained with them are contained in the directory: results. Towards the end each file are inserted the compilation instructions used on the macbook air and on the Asus Optiplex running Ubuntu 18.04.5
Precomputations:
Kummer_main.c: C program.
input: an odd prime q.
output: the ascii file primroot.res contains q and g, a primitive root mod q.
FFT programs : (using the FFTW library). All these programs use the FFTW-guru64 interface (which also allows to transform sequences having dimension larger than 231-1).

Kummer-bernoulli-fftwl.c: C program. It computes the generalised Bernoulli numbers via a sum over a, 1≤a≤q-1, using FFT; it needs the fftw library. It uses the decimation in frequency strategy before calling fftw routines. It's the long double precision version. input: the ascii files primroot.res; output: the value of r(q).

Kummer-psi-fftwl.c: C program. It computes the sum over a, 1≤a≤q-1, of ψ(a/q) using FFT; it needs the fftw library. It uses the decimation in frequency strategy before calling fftw routines. It's the long double precision version. input: the ascii files primroot.res; output: the value r(q). An alternative version which uses the ψ function of PARI/Gp (precomputed and stored on an external file; see my page dedicated to the computation of the Euler-Kronecker constants for prime cyclotomic fields) and one using the GSL ψ function are also available on request together with the quadruple precision versions of all the previously mentioned programs.

Results
The results presented in  can be retrieved as follows.
The ones for q up to 9689 are contained towards the end of the each gp scripts listed before.
The ones for q from 1000003 up to 9854964401 were obtained with the C programs (and the FFTW library) are collected as .zip files in the directory results.
The ones for every prime between 3 and 107 can be found in a csv file here results; the analysis on this file were performed using a python3-pandas script (also included there).
The results of 𝔊q-𝔊q+ (much easier to compute than its summands, see ) for every prime between 3 and 107 can be found in a csv file here results; the analysis on this file were performed using a python3-pandas script (also included there).
In the directory plots you can find the histograms and the scatter plots of the normalised results of 𝔊q-𝔊q+ and r(q) for every prime between 3 and 107.

References

Some of the mathematical papers connected with this computational project are the following.
 A. Languasco - Efficient computation of the Euler-Kronecker constants for prime cyclotomic fields - Research in Number Theory 7 (2021), no. 1, Paper no. 2.
 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.
 A. Languasco, P. Moree, S. Saad Eddin, A. Sedunova - Computation of the Kummer ratio of the class number for prime cyclotomic fields , Arxiv, 2019.
 K. Ford; F. Luca; P. Moree - Values of the Euler phi-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields  - Math. Comp. 83 (2014), 1457-1476.
 P. Moree - Irregular Behaviour of Class Numbers and Euler-Kronecker Constants of Cyclotomic Fields: The Log Log Log Devil at Play, in Irregularities in the Distribution of Prime Numbers. From the Era of Helmut Maier's Matrix Method and Beyond (J. Pintz and M.Th. Rassias, eds.), Springer, 2018, pp. 143-163.
 M.A. Shokrollahi - Relative class number of imaginary abelian fields of prime conductor below 10000 , Math. Comp. 68 (1999), 1717-1728.

Ultimo aggiornamento: 10.12.2021: 15:13:54

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