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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) [2024].
English C1 badge.
Implementation of Pintz-Ruzsa method
for exponential sums over powers of two.
A. Languasco
and
A. Zaccagnini
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
Roughly speaking the problem is the following.
Let L = log2 X and G(α) = Σm≤L e(2mα) where 0<α≤1.
We would like to evaluate the constant v=v(c), 0< v <1, such that
| G(α) | ≤ v L
for every α in (0,1)\ E(v) where
| E(v) | ≤ X− c.
The actual computations were performed using the following software on the NumLab pcs of the Department of Pure and Applied Mathematics of the University of Padova.
Software
Since several people started to use these programs without citing our paper, I decided to remove them from this site. (08/08/2022)
The software is now GPL-licensed and with a DOI: 10.24433/CO.2996876.v2
A Code Ocean capsule (able to run an example of use of such a program) is here:
-
PRmethodfinal.gp:
Pari/GP
script. It can be used via
gp2c.
This version was used in reference [1].
The main function is PintzRuzsa_psiapprox(c,k,numdigits)
Input: c is the level for the set E, k is the degree of the used polynomials, numdigits is the precision for the final result on v
Output: the constant v evaluated with with an error < 10-numdigits
Results-PRmethodfinal: pdf file. Results of PRmethodfinal.gp with numigits = 10, 20, 30, 50. -
PRmethod-KB.gp:
Pari/GP
script. This is an improved (by K. Belabas) version of
the previous script.
This version is about 15% faster for small precisions
and 5% faster for large precisions.
It can be used via
gp2c.
This version was used in reference [1].
The main function is PintzRuzsa_psiapprox(c,k,numdigits)
Input: c is the level for the set E, k is the degree of the used polynomials, numdigits is the precision for the final result on v
Output: the constant v evaluated with an error < 10-numdigits
Results-PRmethod-KB: pdf file. Results of PRmethod-KB.gp with numigits = 10, 20, 30, 50. -
PRmethod-KB-2.gp:
Pari/GP
script. Improved dyadic search in the main function. This lets
us work with inputs very near to 0.
It can be used via gp2c. This version was used in references [4], [5], and in the Ph.D. theses by Settimi and Rossi (listed below). The results of the computation used in [4] is contained at the bottom of the program file.
The main function is PintzRuzsa_psiapprox(c,k,numdigits)
Input: c is the level for the set E, k is the degree of the used polynomials, numdigits is the precision for the final result on v
Output: the constant v evaluated with an error < 10-numdigits
References
The papers connected with this computational project are the following ones together with the references listed there.
[1] A. Languasco, A. Zaccagnini - On a Diophantine problem with two primes and s powers of two - Acta Arithmetica 145 (2010), 193--208
[2] J. Pintz and I.Z. Ruzsa - On Linnik's approximation to Goldbach's problem, I - Acta. Arith., 109:169--194, 2003.
[3] Pari/GP, version 2.3.5, Bordeaux, 2010, http://pari.math.u-bordeaux.fr/
[4] A. Languasco, V. Settimi - On a Diophantine problem with one prime, two squares of primes and s powers of two - Acta Arithmetica, 154 (2012), 385--412, Computational part.
[5] A. Languasco, A. Zaccagnini - A Diophantine problem with a prime and three squares of primes - Journal of Number Theory. 132 (2012), no. 12, 3016--3028. MR , ZBL .
Acknowledgements
We would like to thank Imre Ruzsa for sending us his original U-Basic code for this program and Karim Belabas for helping us in improving the performance of our Pari/GP code for the Pintz-Ruzsa algorithm.
Other researcher's papers
As I expected, it turned out that these values and/or softwares were useful to other researchers; so far they were used in the following papers even if, sometimes, our papers listed before or this webpage are not cited (I don't know why):
Shezad Hathi - Representation of even integers as a sum of squares of primes and powers of two - Acta Arithmetica 206 (2022), 353--372.
Yuhui Liu - Two results on Goldbach-Linnik problems for cubes of primes - Rocky Mountain J. Math. 52 (2022), 999-1007. (The author uses our script but he does not cite our paper [1])
X. Zhao - Goldbach-Linnik type problems on cubes of primes - The Ramanujan Journal 57 (2022), 239-251. (The author uses our script but he does not cite our paper [1])
Y. Wang - Diophantine approximation with two primes and powers of two - The Ramanujan Journal, 39 (2016), 235-345.
D.J. Platt; T. Trudgian - Linnik's approximation to Goldbach's conjecture, and other problems - Journal of Number Theory, 153 (2015), 54-62. (The authors use our script but they do not cite our paper [1])
Z. Liu; H. Sun - Diophantine Approximation with 4 Squares of Primes and Powers of 2 - Chinese Journal of Contemporary Mathematics, 34 (2013), 361-368.
A. Rossi - The Goldbach-Linnik Problem: Some conditional results - PhD Thesis, Università of Milano, 2011.
V. Settimi - On some additive problems with primes and powers of a fixed integer - PhD Thesis, Università of Padova, 2011.
Changes in this page:
Aug. 16th 2016: added section about other researcher's papers.
Sept 12th, 2020: updated section about other researcher's papers.
Sept 28th, 2021: more updates in the section about other researcher's papers.
March 08th, 2022: more updates in the section about other researcher's papers.
Ultimo aggiornamento: 28.09.2024: 11:32:14
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