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Small values of  L'/L (1, χ) 
A. Languasco
(for a paper in collaboration with Youness Lamzouri)
In a recent paper [2], I introduced a fast method to compute the
values of L'/L (1, χ), where χ is a Dirichlet character with prime
modulus. In a joint effort with Y. Lamzouri, see [1], we then studied,
both theoretically and computationally, the size of m_{q }, the minimum
over non principal characters
for  L'/L (1, χ) .
In the paper [1], coauthored with Y. Lamzouri, we proved an upper bound for this quantity
and we had computationally evaluate its behaviour for every
modulus q (q odd prime up to 10^{7}).
We also formulate a conjecture about the order of magnitude
of m_{q }.
In this page we collect some links concerning such a computation.
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:
These scripts were developed using PARI/GP v. 2.11.2, but their correctness
were subsequently verified up to v. 2.13.0.
Mindirectfinal.gp:
PARI/GP
script. It can be used via
gp2c.
The function to be run is:
min_direct (r_{1},r_{2},defaultprecision).
Input: 2< r_{1} < r_{2}, two integers; defaultprecision is the number of digits requested.
Output: the value m(q) (distinguishing between even and odd characters)
for every odd prime q such that r_{1}≤q≤r_{2} and the running times.
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.
MinSfinal.gp:
PARI/GP
script. It can be used via
gp2c.
The function to be run is:
minS_DIF (r_{1},r_{2},defaultprecision).
Input: 2< r_{1} < r_{2}, two integers; defaultprecision is the number of digits requested.
Output: the value m(q) (distinguishing between even and odd characters)
for every odd prime q such that r_{1}≤q≤r_{2} and the running times.
Comments: it computes first the values of log(Γ) and the decimated in frequency
values of S at the a/q points, see [2], and then obtain m(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.
MinTfinal.gp:
PARI/GP
script. It can be used via
gp2c.
The function to be run is:
minT (r_{1},r_{2},defaultprecision).
Input: 2< r_{1} < r_{2}, two integers; defaultprecision is the number of digits requested.
Output: the value m(q) (distinguishing between even and odd characters)
for every odd prime q such that r_{1}≤q≤r_{2} and the running times.
Comments: it computes first the needed values of ψ and T at the a/q points,
see [2], and then obtain m(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
MinSfftwl.c:
C program.
It computes m(q) via FFT; it needs the fftw library.
It's the long double precision version. Values of log(Γ) are computed
using the internal function of the C programming language.
It uses the decimated in frequency values for S and the sequence g^{k} mod q.
input: the ascii files primroot.res, precomp_SDIF.res.
output: the value m(q) (distinguishing between even and odd characters) and the running times.
MinSfftwq.c:
C program.
It computes m(q) via FFT; it needs the fftw library.
It's the quadruple precision version. Values of log(Γ) are computed
using the internal function of the C programming language.
It uses the decimated in frequency values for S and the sequence g^{k} mod q.
input: the ascii files primroot.res, precomp_SDIF.res.
output: the value m(q) (distinguishing between even and odd characters) and the running times.
MinTfftwl.c:
C program.
It computes m(q) via FFT; it needs the fftw library.
It's the long double precision version.
It uses the values for T, ψ and the sequence g^{k} mod q.
Double precision values of ψ are computed
using the GSL library.
input: the ascii files primroot.res, precomp_T.res.
output: the value m(q) (distinguishing between even and odd characters) and the running times.
Results
The results presented in [1] can be retrieved as follows.
The results for m_{q}
for every prime between 3 and 1000 are contained at
the bottom of each gp script.
The results for m_{q}
for every prime between 3 and 10^{6} were obtained
starting with the values of S(a/q) computed in [2], with the
C programs (and the FFTW library) on a Dell Optiplex machine
(Intel i57500 processor, 3.40GHz,
16 GB of RAM and running Ubuntu 18.04.2).
The results for m_{q}
for every prime between 10^{6} and 10^{7} were obtained
making use of a new algorithm for computing S(a/q)
presented in [3].
All these results can be found in a csv file here:
results.
The analysis on this file were performed using a python3pandas
script (also included there).
In particular, in the directory
results/P6119053
you will find the output of the program presented in [3],
suitable modified to get the indexes
needed to identify the characters that gave rise to
the minimal (and maximal) values. This is meaningful
since min(m_{q}), 3 < q < 10^{7}, is attained at q=6119053.
Look at the file named P6119053charminmaxdetection.txt and look
for the "min_odd(6119053)" and "chiminodd index" strings; the notation
used to identify the character is explained on sect. 5.2 of [1]
(or on sect 4.1 of [2]).
In the directory
plots you can find the scatter
plots of m_{q} and m_{q}' = 200/21 * q* m_{q}
for every prime between 3 and 10^{7}
(used in [1]).
References
Some of the mathematical papers connected with this project are the following.
[1]
Y. Lamzouri, A. Languasco 
Small values of  L'/L (1, χ) 
,
Experimental Mathematics,
electronically published on September 3, 2021,
(to appear in print).
[2] A. Languasco 
Efficient computation of the EulerKronecker 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 RamanujanDeninger
gamma function and some numbertheoretic applications
 Mathematics of Computation 90 (2021), 28992921.
Ultimo aggiornamento: 10.12.2021: 15:14:26
