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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 [3], I (A. Languasco) compared three different methods to compute the
Kummer ratio r(q) for prime cyclotomic fields.
In particular in [3] 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 [3])
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:
Howitworks.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.LanguascoPrograms
PARI/Gp scripts
Kummerdirectfinal.gp:
PARI/GP
script. It can be used via
gp2c.
The function to be run is:
global_kummer_direct (r_{1},r_{2},defaultprecision).
Input: 2< r_{1} < r_{2}, two integers; defaultprecision is the number of digits requested.
Output: r(q) for every odd prime q such that r_{1}≤q≤r_{2}.
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.
Kummerpsifinal.gp:
PARI/GP
script. It can be used via
gp2c.
The function to be run is:
global_kummer_psi (r_{1},r_{2},defaultprecision).
Input: 2< r_{1} < r_{2}, two integers; defaultprecision is the number of digits requested.
Output: r(q) for every odd prime q such that r_{1}≤q≤r_{2}.
Comments: it computes first the needed values of ψ at the a/q points, see [3],
and then obtain r(q) with a trivial implementation of the sum over a, 1≤a≤q1.
Examples on how to use the function
and computational results are collected towards the end of the file.
KummerBernoullifinal.gp:
PARI/GP
script. It can be used via
gp2c.
The function to be run is:
global_kummer_bernoulli (r_{1},r_{2},defaultprecision).
Input: 2< r_{1} < r_{2}, two integers; defaultprecision is the number of digits requested.
Output: r(q) for every odd prime q such that r_{1}≤q≤r_{2}.
Comments: it computes the sequence g^{k} mod q, where g is a primitive root of
Z_{q}^{*}.
Then it obtains the generalised Bernoulli numbers and hence r(q) with a trivial
implementation of the sum over a, 1≤a≤q1. 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 FFTWguru64 interface (which also allows to transform
sequences having dimension larger
than 2^{31}1).
Kummerbernoullifftwl.c:
C program. It computes the generalised Bernoulli numbers via a sum over a, 1≤a≤q1,
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).
Kummerpsifftwl.c:
C program.
It computes the sum over a, 1≤a≤q1, 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 EulerKronecker 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 [3] 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 10^{7}
can be found in a csv file here results;
the analysis on this file were performed using a python3pandas
script (also included there).
The results of 𝔊_{q}𝔊_{q}^{+}
(much easier to compute than its summands, see [1]) for every
prime between 3 and 10^{7}
can be found in a csv file here results;
the analysis on this file were performed using a python3pandas
script (also included there).
In the
directory plots you can find the histograms and the scatter
plots of the normalised results of 𝔊_{q}𝔊_{q}^{+}
and r(q) for every prime between 3 and 10^{7}.
References
Some of the mathematical papers connected with this computational project are the following.
[1] A. Languasco 
Efficient computation of the EulerKronecker constants for prime cyclotomic fields
 Research in Number Theory 7 (2021), no. 1, Paper no. 2.
[2] A. Languasco, L. Righi 
A fast
algorithm to compute the RamanujanDeninger
gamma function and some numbertheoretic applications
 Mathematics of Computation 90 (2021), 28992921.
[3] A. Languasco, P. Moree, S. Saad Eddin, A. Sedunova 
Computation of the Kummer ratio of the class number for prime cyclotomic fields
, Arxiv, 2019.
[4] K. Ford; F. Luca; P. Moree 
Values of the Euler phifunction not divisible by a given odd prime,
and the distribution of EulerKronecker constants for cyclotomic fields
 Math. Comp. 83 (2014), 14571476.
[5] P. Moree 
Irregular Behaviour of Class Numbers and EulerKronecker
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. 143163.
[5]
M.A. Shokrollahi 
Relative class number of imaginary abelian fields of
prime conductor below 10000
, Math. Comp. 68 (1999), 17171728.
Ultimo aggiornamento: 10.12.2021: 15:13:54
