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.

Small values of | L'/L (1, χ) | A. Languasco (for a paper in collaboration with Youness Lamzouri)

In a recent paper , 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 , we then studied, both theoretically and computationally, the size of mq , the minimum over non principal characters for | L'/L (1, χ) |. In the paper , co-authored 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 107). We also formulate a conjecture about the order of magnitude of mq . 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.Languasco-Programs

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.
Min-direct-final.gp: PARI/GP script. It can be used via gp2c. The function to be run is:
min_direct (r1,r2,defaultprecision).
Input: 2< r1 < r2, 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 r1≤q≤r2 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.
Min-S-final.gp: PARI/GP script. It can be used via gp2c. The function to be run is:
minS_DIF (r1,r2,defaultprecision).
Input: 2< r1 < r2, 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 r1≤q≤r2 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 , and then obtain m(q) with a trivial implementation of the sum over a, 1≤a≤q-1. Examples on how to use the function and computational results are collected towards the end of the file.
Min-T-final.gp: PARI/GP script. It can be used via gp2c. The function to be run is:
minT (r1,r2,defaultprecision).
Input: 2< r1 < r2, 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 r1≤q≤r2 and the running times.
Comments: it computes first the needed values of ψ and T at the a/q points, see , and then obtain m(q) with a trivial implementation of the sum over a, 1≤a≤q-1. Examples on how to use the function and computational results are collected towards the end of the file.

C programs
Min-S-fftwl.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 gk mod q.
input: the ascii files primroot.res, precomp_S-DIF.res.
output: the value m(q) (distinguishing between even and odd characters) and the running times.
Min-S-fftwq.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 gk mod q.
input: the ascii files primroot.res, precomp_S-DIF.res.
output: the value m(q) (distinguishing between even and odd characters) and the running times.
Min-T-fftwl.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 gk 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  can be retrieved as follows.
The results for mq for every prime between 3 and 1000 are contained at the bottom of each gp script.
The results for mq for every prime between 3 and 106 were obtained starting with the values of S(a/q) computed in , with the C programs (and the FFTW library) on a Dell Optiplex machine (Intel i5-7500 processor, 3.40GHz, 16 GB of RAM and running Ubuntu 18.04.2).
The results for mq for every prime between 106 and 107 were obtained making use of a new algorithm for computing S(a/q) presented in .
All these results can be found in a csv file here: results. The analysis on this file were performed using a python3-pandas script (also included there).
In particular, in the directory results/P6119053 you will find the output of the program presented in , 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(mq), 3 < q < 107, 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  (or on sect 4.1 of ).
In the directory plots you can find the scatter plots of mq and mq' = 200/21 * q* mq for every prime between 3 and 107 (used in ).

References

Some of the mathematical papers connected with this project are the following.
 Y. Lamzouri, A. Languasco - Small values of | L'/L (1, χ) | , Experimental Mathematics, electronically published on September 3, 2021, (to appear in print).
 A. Languasco - Efficient computation of the Euler-Kronecker constants for prime cyclotomic fields - Research in Number Theory 7 (2021), no. 1, Paper no. 2.
 A. Languasco, L. Righi - A fast algorithm to compute the Ramanujan-Deninger gamma function and some number-theoretic applications - Mathematics of Computation 90 (2021), 2899--2921.

Ultimo aggiornamento: 10.12.2021: 15:14:26

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