Alessandro Languasco

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Alessandro Languasco


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 non-principal 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 107; using such data we verified, for q in such a range, Littlewood's estimates in [3] and the Lamzouri-Li-Soundararajan [1] effective bounds.
In the paper [3], co-authored with T. Trudgian, we improved the estimates in [1] proving that they hold for every non-principal 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.Languasco-Programs



PARI/Gp scripts
MaxminL-composite.gp: PARI/GP script. It can be used via gp2c. The function to be run is:
maxminL_comp (r1,r2,defaultprecision).
Input: 2< r1 < r2, two integers; defaultprecision is the number of digits requested.
Output: the value Mq and mq for every composite integer q such that r1≤q≤r2 and the running times. It saves the results ond the files maxL-values-comp.csv, minL-values-comp.csv, maxminL-times-comp.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.
MaxminL-all.gp: PARI/GP script. It can be used via gp2c. The function to be run is:
maxminL_all (r1,r2,defaultprecision).
Input: 2< r1 < r2, two integers; defaultprecision is the number of digits requested.
Output: the value Mq and mq for every integer q such that r1≤q≤r2 and the running times. It saves the results ond the files maxL-values-all.csv, minL-values-all.csv, maxminL-times-all.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 mq and Mq for every q between 3 and 1000 are contained in the files maxL-values-all.csv, minL-values-all.csv here.
In the directory plots you can find the scatter plots of mq and Mq 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: python3-pandas scripts.

To verify the inequalities on Mq: run the script named analysis-MaxL.py on the numerical results contained in maxL-values.csv (renamed version of maxL-values-all.csv); the output file named analysis-maxL.txt contains the information to verify the inequalities on Mq.
To verify the inequalities on mq: run the script named analysis-MinL.py on the numerical results contained in minL-values.csv (renamed version of minL-values-all.csv); the output file named analysis-minL.txt contains the information to verify the inequalities on mq.
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 non-residue and related problems, Math. Comp. 84 (2015), 2391--2412. Corrigendum ibid., Math. Comp. 86 (2017), 2551--2554.
[2] A. Languasco - Numerical verification of Littlewood's bounds for |L(1,χ)| , Journal of Number Theory 223 (2021), 12--34. 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), 358--372.



Ultimo aggiornamento: 10.12.2021: 15:13:25

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