
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.
Implementation of PintzRuzsa method
for exponential sums over powers of two.
A. Languasco
and
A. Zaccagnini
In this page we collect some programs and results
concerning the computation of upper bounds for
exponential sums over powers of two.
See the paper by PintzRuzsa [2] to see the definitions
of the main functions and parameters.
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
Roughly speaking the problem is the following.
Let
L = log_{2} X and
G(α) = Σ_{m≤L} e(2^{m}α)
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

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}
ResultsPRmethodfinal:
pdf file. Results of PRmethodfinal.gp with numigits = 10, 20, 30, 50.

PRmethodKB.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}
ResultsPRmethodKB:
pdf file. Results of PRmethodKB.gp with numigits = 10, 20, 30, 50.

PRmethodKB2.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), 193208
[2]
J. Pintz and I.Z. Ruzsa 
On Linnik's approximation to Goldbach's problem, I

Acta. Arith., 109:169194, 2003.
[3] PARI/GP, version 2.3.5, Bordeaux, 2010,
http://pari.math.ubordeaux.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), 385412,
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, 30163028.
MR ,
ZBL .
Acknowledgements
We would like to thank Imre Ruzsa
for sending us his original UBasic code for
this program
and Karim Belabas for helping us in improving the performance
of our PARI/GP code for the PintzRuzsa 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):
Yuhui Liu 
Two results on GoldbachLinnik problems for cubes of primes
 Rocky Mountains Journal of Mathematics,
to appear (2021).
(The author uses our script but he does not cite our paper [1] !!)
X. Zhao 
GoldbachLinnik type problems on cubes of primes
 The Ramanujan Journal,
(2020).
(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), 235345.
D.J. Platt; T. Trudgian 
Linnik's approximation to Goldbach's conjecture, and other problems
 Journal of Number Theory,
153
(2015), 5462.
(The authors use our script but the 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), 361368.
A. Rossi 
The GoldbachLinnik 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.
Ultimo aggiornamento: 10.12.2021: 15:13:48
