
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.
Uniform effective estimates for L(1,χ)
A. Languasco
(for a paper in collaboration with Timothy S. Trudgian)
In this page I collect some links concerning the computation of the
values of L(1,χ), where χ is a nonprincipal Dirichlet character mod q.
In my paper [2], I introduced a fast algorithm
to compute the values of L(1,χ) which is based on a FFT strategy.
I was able to evaluate such functions for every odd prime q up to 10^{7};
using such data we verified, for q in such a range, Littlewood's estimates in [3]
and the LamzouriLiSoundararajan [1] effective bounds.
In the paper [3], coauthored with T. Trudgian, we improved the estimates in [1] proving that
they hold for every nonprincipal primitive Dirichlet characters χ mod q,
q >= 404.
In this page I show how to compute L(1,χ)
for every q, 3<=q<=1000. The results for primes q are also contained
in this page
Littlewood_ineq.html.
I describe here the Pari/Gp scripts used to achieve these results.
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
MaxminLcomposite.gp:
PARI/GP
script. It can be used via
gp2c.
The function to be run is:
maxminL_comp (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} and m_{q}
for every composite integer q such that r_{1}≤q≤r_{2} and the running times.
It saves the results ond the files maxLvaluescomp.csv, minLvaluescomp.csv, maxminLtimescomp.csv.
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.
MaxminLall.gp:
PARI/GP
script. It can be used via
gp2c.
The function to be run is:
maxminL_all (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} and m_{q}
for every integer q such that r_{1}≤q≤r_{2} and the running times.
It saves the results ond the files maxLvaluesall.csv, minLvaluesall.csv, maxminLtimesall.csv.
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.
Numerical results
The numerical results presented in [3] can be retrieved as follows.
The results for m_{q} and M_{q}
for every q between 3 and 1000 are contained
in the files maxLvaluesall.csv, minLvaluesall.csv
here.
In the directory
plots you can find the scatter
plots of m_{q} and M_{q} for every integer between 3 and 1000.
Proofs of the inequalities presented in [3] for 3<=q <= 1000
The verification of the inequalities presented in [3]
uses two python3 scripts on the numerical results previously
mentioned.
They can be downloaded here:
python3pandas scripts.
To verify the inequalities on M_{q}:
run the script named analysisMaxL.py on the numerical results
contained in maxLvalues.csv (renamed version of maxLvaluesall.csv); the output file named analysismaxL.txt
contains the information to verify the inequalities on M_{q}.
To verify the inequalities on m_{q}:
run the script named analysisMinL.py on the numerical results
contained in minLvalues.csv (renamed version of minLvaluesall.csv); the output file named analysisminL.txt
contains the information to verify the inequalities on m_{q}.
References
Some of the mathematical papers connected with this project are the following.
[1]
Y. Lamzouri, X. Li, K. Soundararajan,
Conditional bounds for the least quadratic nonresidue and related problems, Math. Comp.
84 (2015), 23912412. Corrigendum ibid.,
Math. Comp. 86 (2017), 25512554.
[2] A. Languasco 
Numerical verification of Littlewood's bounds for L(1,χ)
, Journal of Number Theory 223 (2021), 1234.
Code Ocean capsule
[3]
A. Languasco, T.S. Trudgian 
Uniform effective estimates for  L (1, χ) 
 Journal of Number Theory, electronically published on August
24, 2021, (to appear in print).
[4] J. E. Littlewood,
On the class number of the corpus P(sqrt{k}), Proc.
London Math. Soc. 27 (1928), 358372.
Ultimo aggiornamento: 10.12.2021: 15:13:25
