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Computation of Ramanujan-Deninger Gamma function
and some number-theoretic applications
A. Languasco and L. Righi
In this page we collect some links concerning the computation of the Ramanujan-Deninger Gamma function and some of its number theoretic applications (mainly about the Euler-Kronecker constants for prime cyclotomic fields and the maximum over non trivial characters for the logarithmic derivative at 1 of Dirichlet L-functions modulo q, q being odd prime up to 107).
In a recent paper [1], we introduced a fast algorithm to compute a generalisation of Euler's Gamma function that we named Ramanujan-Deninger Gamma function since many theoretical results are contained in the second Ramanujan's notebook and in a paper of Deninger.
Combining such an algorithm with the FFT strategy used in [2] we were able to obtain that
Recalling that the first negative value for 𝔊q was obtained in 2014 by Ford, Luca and Moree in [3] (𝔊964477901 = -0.182374...) and that other two examples, 𝔊9109334831 = -0.248739... and 𝔊9854964401 = -0.096465..., are listed in [2], see also this page, we see that 50040955631 is 50-times larger than the case provided by Ford, Luca and Moree and about 5-time larger than the ones in [2].
The fact of being able to find such a large q with 𝔊q<0 depends on the use of the algorithm described in [1] to compute S.
Here we describe here the gp scripts and C programs we developed and 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
DIFprecS-new-v5.gp: Pari/GP script. It can be used via gp2c. The function to be run is: DIFprecS_new(x,y).
Input: x,y two integers; the ascii file primroot.res containing a prime q and g, a primitive root mod q.
Output: it computes -(S(a/q)+S(1-a/q)) with the algorithm described in [1], a =gj mod q, g=primroot, 1-a/q = gj+m/q, m=(q-1)/2, for j in a given interval [x,y]. It saves the results on an ASCII file.
precT-new-v3.gp: Pari/GP script. It can be used via gp2c. The function to be run is: precT_new(x,y).
Input: x,y two integers; the ascii file primroot.res containing a prime q and g, a primitive root mod q.
Output: it computes -T(a/q) with the algorithm described in [1], a =gj mod q, g=primroot, for j in a given interval [x,y]. It saves the results on an ASCII file.
C programs: Euler-Kronecker constants computation:
EKmain.c: C program.
input: an odd prime q.
output: the ascii file primroot.res contains q and g, a primitive root mod q.
Steps-norms-final.zip: zip archive containing several C programs that
- using the algorithm described in [1], it computes the needed decimated in frequency values of -(S(a/q)+S(1-a/q)), a =gj mod q, g=primroot, 1-a/q = gj+m/q, m=(q-1)/2, for j in a given interval [x,y]. Saves the results on an ASCII or binary file;
- uses such values to compute 𝔊q+, 𝔊q and Mq via FFT using the algorithm described in [1] and [2];
- the programs use the FFTW-guru64 interface (which also allows to transform sequences having dimension larger than 231-1). Moreover we modified the FFTW internal way of storing objects in memory to be able to store them directly on the hard disks; to achieve this we inserted the mmap() UNIX system call in the body of FFTW code;
- the standard precision is 80 bits (long double precision of the C programming language);
- evaluates at run-time the accuracy of the FFT computations.
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).
Results
The results presented in [1] and [2] can be retrieved as follows.
The results for 𝔊q+, 𝔊q, Mq (computed in [2]) for every prime between 3 and 106 were obtained with the C programs (and the FFTW library) on the cluster of the Math. Dept. of the University of Padova; they 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 the directory plots you can find the scatter plots of the normalised results of 𝔊q+, 𝔊q, Mq for every prime between 3 and 106.
The results for the same quantities for every prime between 106 and 107 (computed in [1]) are contained in a csv file here: results; the analysis on this file were performed using a python3-pandas script (also included there). In the directory plots you can find the scatter plots of the normalised results of 𝔊q+, 𝔊q, Mq for every prime between 106 and 107.
The results for 𝔊q+ and 𝔊q with q=50040955631 were obtained with the C programs (and the FFTW library) on the CAPRI (Calcolo ad Alte Prestazioni per la Ricerca e l'Innovazione) infrastructure of the University of Padova.
References
Some of the papers connected with this project are the following.
[1] 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.
[2] A. Languasco - Efficient computation of the Euler-Kronecker constants for prime cyclotomic fields - Research in Number Theory 7 (2021), no. 1, Paper no. 2.
[3] K. Ford; F. Luca; P. Moree - Values of the Euler phi-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields - Math. Comp. 83 (2014), 1457-1476.
[4] P. Moree - Irregular Behaviour of Class Numbers and Euler-Kronecker Constants of Cyclotomic Fields: The Log Log Log Devil at Play, in Irregularities in the Distribution of Prime Numbers. From the Era of Helmut Maier's Matrix Method and Beyond (J. Pintz and M.Th. Rassias, eds.), Springer, 2018, pp. 143-163.
Ultimo aggiornamento: 28.09.2024: 10:48:27
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