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## Programs and numerical results for the paper ``Landau and Ramanujan approximations for divisor sums and coefficients of cusp forms'' by A. Ciolan, A. Languasco and P. Moree

In this page I (A. Languasco) include my programs (Pari/Gp and Python scripts) developed to obtain the numerical results described in the paper [1], co-authored with Alexandru Ciolan and Pieter Moree.
For the definition of the quantities γq,k, γ'q,k, γKr, γK2r and S(r,q), please refer to [1].
In the following the acronym ``LvR'' stands for the Landau versus Ramanujan problem as stated in Section 1.2.2 of [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 and Python scripts
gammaK.gp: Pari/Gp script. It can be used via gp2c. The function to be run is: gamma_K(q1, q2, prec).
Input: q1, q2, prec: three positive integers.
Output: it computes the Euler-Kronecker constants γKr and γK2r of the cyclotomic subfields Kr and K2r for every r | (q-1)/2, 1 ≤ r ≤ 6; where q is an odd prime running between q1 and q2. The computation is performed with an accuracy of prec decimal digits. It uses the algorithm developed in [2]-[3] for computing the Euler-Kronecker constant of a cyclotomic field modified as in Section 9 of [1] to be able to handle the case of such cyclotomic subfields.
The output is saved in one file for each 1 ≤ r ≤ 6 for further elaborations needed to study the LvR problem, see the Python script below.
In the folder gammaK-results you'll find the result of a computation performed with q1 = 3; q2 = 3000 and prec = 30. Each file contains the results according the values of r, 1 ≤ r ≤ 6.
Srall-v2.gp: Pari/Gp script. It can be used via gp2c. The function to be run is: Srall(r1, r2, q1, q2, Pbound, prec).
Input: r1, r2, q1, q2, Pbound, prec: six positive integers.
Output: it computes −S(r,q) (please remark the change of sign), with 1 ≤ r1 ≤ r ≤ r2 ≤ 6; q1 ≤ q ≤ q2 ; q is an odd prime, by truncating up to Pbound the sums in its definition; prec is the internal decimal precision used.
The output is saved in one file for each 1 ≤ r ≤ 6 for further elaborations needed to study the LvR problem, see the Python script below.
In the folder Sr-values-results you'll find the result of a computation of S(r,q) performed with 1 ≤ r ≤ 6. Pbound can be 108, 109 or 1010 (with prec = 19); see Section 9 of [1]. The results were first computed with Pbound = 108; the ones not having a sufficiently good accuracy were recomputed with Pbound = 109; the ones not having (yet) a sufficiently good accuracy were recomputed with Pbound = 1010. All these results were then merged in the files mentioned before.
analysis-gen-gammakq.py: Python script; it uses pandas and numpy. It computes lower and upper bounds for γk,q and γ'k,q; then it decides on the LvR problem, see Theorem 4 and Conjectures 1-2 in [1].
Input: it needs in the folder "inputs" the output files of the gammaK.gp and Srall.gp scripts described above.
Output: in the folder "outputs" writes the file LvR-r=*.csv containing the LvR-analysis for every 1 ≤ r ≤ 6.
In the folder LvR you'll find the results described in Theorem 4 and Conjectures 1-2 of [1].
gamma(q=2).gp: Pari/Gp script. It can be used via gp2c. The function to be run is: gamma2(Pbound, prec).
Input: Pbound, prec: two positive integers.
Output: it computes γ1,2 and γ'1,2 by truncating up to Pbound the sum in their definitions, see Section 3.7 of [1]. prec is the internal decimal precision used.
Results are collected towards the bottom of the script.
ShanksB-v3.gp: Pari/Gp script. It can be used via gp2c. The function to be run is: shanks(prec).
Input: prec: a positive integer.
Output: it computes the Landau-Ramanujan K, Shanks c, and Moree γSB constants using Shanks' acceleration technique, see Section 9.3 of [1]. As a byproduct, it also computes the Catalan constant.
Results are collected towards the bottom of the script.
Th2-Th4-Remark15-numerics.gp: Pari/Gp script. It can be used via gp2c. It collects the functions needed to perform the numerical verification needed in the proofs of Theorems 2, 4 and in Remark 15 of [1].
Results are collected towards the bottom of the script.
gammaT_2331.gp: Pari/Gp script. It can be used via gp2c. It computes γT for the modular forms of weight 12 (modulo 23) and 16 (modulo 31) using the results of Section 4.4 of [1]. The function to be run is: gammaT_2331(Pbound, prec).
Input: Pbound, prec: two positive integers.
Output: it computes γT for the modular forms of weight 12 and 16 (respectively modulo 23 and 31) by truncating up to Pbound the sums in their definitions; prec is the internal decimal precision used.
Results are collected towards the bottom of the script.
gamma_quadratic-v2.gp: Pari/Gp script. It can be used via gp2c. It computes γ(q-1)/2,q of the quadratic field case described Section 3.10 of [1]. In this case we can use Shanks' acceleration technique. The function to be run is: gamma_quad(q1 ,q2, prec).
Input: q1, q2, prec: three positive integers.
Output: it computes γ(q-1)/2,q of the quadratic field case for each odd prime q from q1 to q2. It uses eq. (45) of [1] and Shanks' acceleration technique on the involved prime sums; prec is the internal decimal precision used.
Results of a computation with q1 = 3, q2 = 3000 and prec = 50 are collected towards the bottom of the script and here.
Num-Obs1.txt: Text file containing a report about the verification described in Numerical Observation 1 of [1] (performed with Pari/Gp).

Results
The results presented in [1] can be retrieved as follows.
The numerics needed to settle the LvR problem in Theorem 4 and used in Conjectures 1-2 are collected in the folder LvR.
The numerics needed in the Tables contained in [1] are collected here: tables.
The numerical data required in Lemma 8 of [1] are available here: checkLemma8.

References

Some of the mathematical papers connected with this project are the following.
[1] A. Ciolan, A. Languasco and P. Moree - Landau and Ramanujan approximations for divisor sums and coefficients of cusp forms - ArXiv, 2021.
[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] 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: 15:12:33

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