
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.
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],
coauthored with Alexandru Ciolan and Pieter Moree.
For the definition of the quantities γ_{q,k}, γ'_{q,k},
γ_{Kr},
γ_{K2r} 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.LanguascoPrograms
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 EulerKronecker constants γ_{Kr}
and γ_{K2r} of the cyclotomic subfields K_{r}
and K_{2r}
for every r  (q1)/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 EulerKronecker 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 gammaKresults
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.
Srallv2.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 Srvaluesresults
you'll find the result of a computation of S(r,q) performed with 1 ≤ r ≤ 6.
Pbound can be 10^{8}, 10^{9} or 10^{10} (with prec = 19);
see Section 9 of [1]. The results were first computed with Pbound = 10^{8};
the ones not having a sufficiently good accuracy were recomputed with Pbound = 10^{9};
the ones not having (yet) a sufficiently good accuracy were recomputed with Pbound = 10^{10}.
All these results were then merged in the files mentioned before.
analysisgengammakq.py:
Python script; it uses pandas and numpy.
It computes lower and upper bounds for γ_{k,q} and γ'_{k,q};
then it decides on the LvR problem, see Theorem 4
and Conjectures 12 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 LvRr=*.csv
containing the LvRanalysis for every 1 ≤ r ≤ 6.
In the folder LvR
you'll find the results described in Theorem 4
and Conjectures 12 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 γ_{1,2} 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.
ShanksBv3.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 LandauRamanujan K, Shanks c, and Moree
γ_{SB} 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.
Th2Th4Remark15numerics.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 γ_{T} 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 γ_{T} 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_quadraticv2.gp:
Pari/Gp
script. It can be used via
gp2c.
It computes γ_{(q1)/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_{(q1)/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.
NumObs1.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 12 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 EulerKronecker 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 RamanujanDeninger
gamma function and some numbertheoretic applications
 Mathematics of Computation 90 (2021), 28992921.
Ultimo aggiornamento: 10.12.2021: 15:12:33
