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Papers:
List Papers;
(with Abstracts);
Curriculum (in Italian):
long version ;
short version;
in English:
long version.
Google Scholar profile.
ResearchGate page.
Orcid ID.
Scopus Author ID.
Web of Science Researcher ID,
Mathematical Reviews page,
Zentralblatt page,
IRIS-CINECA bibliometric parameters (italian ASN) [2023].
English C1 badge.
Programs and numerical results for the paper
``Numerical estimates on the Landau-Siegel zero
and other related quantities''
by Alessandro Languasco
In this page I include my programs (Pari/GP and Python scripts and a C programs) developed to obtain the numerical results described in the paper [1].
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, Python scripts and C programs
LandauSiegel.gp: Pari/GP script. It can be used via gp2c. The function to be run is: LS(q1, q2, prec).
Input: 3 ≤ q1, q2, prec: three positive integers.
Output: it computes the L(1,χ□), c1(q), c2(q), c3(q), c4(q) values (for their definitions, see [1]), and the estimate for β (the Landau-Siegel zero) for every prime q, 3 ≤ q1 ≤ q ≤ q2. The computation is performed with an accuracy of prec decimal digits. It uses the lfun-command of Pari/GP to define the relevant L-function.
The output is saved in a .csv file for further elaborations, see the Python script below.
In the file gp-Lvalues-3-10e6.csv you'll find the result of a computation performed with 3 ≤ q1 = 3; q2 = 106 and an accuracy of 38 decimal digits.
csum_estim.gp: Pari/GP script. It can be used via gp2c. The function to be run is: csum_estim(q1, q2, prec).
Input: 3 ≤ q1, q2, prec: three positive integers.
Output: it computes the c2(q)( = c3(q)+c4(q)), c3(q), c4(q) values (for their definitions, see [1]) for every prime q, 3 ≤ q1 ≤ q ≤ q2.
The output is saved in a .csv file for further elaborations, see the Python script below.
In the file c3-c4-3-10e7.csv you'll find the computed values of c2(q), c3(q), c4(q) for every odd prime q ≤ 107 with an accuracy of 38 decimal digits.
steps-norms-quadchar.zip: zip archive containing several C programs that
- computes L(1,χ□) via the FFT-algorithm using the procedure described in [1]; (in fact it computes several other quantities that either I already used or I plan to report onto in the future);
- the programs use the FFTW-guru64 interface;
- the standard precision is 80 bits (long double precision of the C programming language);
- evaluates at run-time the accuracy of the FFT computations (for more on this, see the relevant section in [1]).
To use such programs, first compile them with the make command and then execute the runSteps.exe program (see the instructions with --usage and --describe).
This is the program used to compute the values of L(1,χ□) for 106 < q ≤ 107, q prime.
analysis-Lquad.py: Python script; it uses the packages pandas, numpy and matplotlib.
It computes the lower and upper bounds for c1(q), c2(q), the estimates for beta, if the values of L(1,χ□) verify Joshi's bounds, the values of the Upper Littlewood and Lower Littlewood indices (ULI and LLI, respectively) and the computed values of the class number of the quadratic field ℚ(√(-q)). These are the values reported in the paper [1].
Input: it needs the file common_code.py, and the input files merged-graph6-Lquad-total.result (contains the merged output of LandauSiegel.gp for q ≤ 106 with the one of the C program for 106 < q ≤ 107) and c3-c4-3-10e7.csv of csum_estim.gp.
Output: it writes the file table-results-10-digits.csv that contains all the data mentioned before and the files Joshifirst-true-results-10-digits.csv, Joshisecond-true-results-10-digits.csv containing the list of q such that Joshi's first and second bounds hold. [REMARK: the last digits might be rounded by the python printing/saving routine]
The file output_analysis_Lquad.txt is its execution printout.
Numerical results
All the numerical results presented in [1], and mentioned in the previous section, can be retrieved in the folder results. [REMARK: the last digits might be rounded by the python printing/saving routine]
In the directory plots you can find the scatter plots of c1(q) and c2(q) for every prime between 3 and 107. In the same folder the corresponding histograms are also collected.
References
Some of the papers connected with this project are the following.
[1] A. Languasco - Numerical estimates on the Landau-Siegel zero and other related quantities - J. Number Theory 251 (2023), 185--209.
[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, L. Righi - A fast algorithm to compute the Ramanujan-Deninger gamma function and some number-theoretic applications - Math. Comp. 90 (2021), 2899--2921.
[4] A. Languasco, T.S. Trudgian - Uniform effective estimates for | L (1, χ) | - J. Number Theory 236 (2022), 245--260.
Ultimo aggiornamento: 15.09.2023: 06:07:16
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