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Implementation of Pintz-Ruzsa 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 Pintz-Ruzsa  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.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
1. PRmethodfinal.gp: PARI/GP script. It can be used via gp2c. This version was used in reference .
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.

2. 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 .
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.

3. 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 , , and in the Ph.D. theses by Settimi and Rossi (listed below). The results of the computation used in  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.
 A. Languasco, A. Zaccagnini - On a Diophantine problem with two primes and s powers of two - Acta Arithmetica 145 (2010), 193--208
 J. Pintz and I.Z. Ruzsa - On Linnik's approximation to Goldbach's problem, I - Acta. Arith., 109:169--194, 2003.
 PARI/GP, version 2.3.5, Bordeaux, 2010, http://pari.math.u-bordeaux.fr/
 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.
 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):
Yuhui Liu - Two results on Goldbach-Linnik 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  !!)
X. Zhao - Goldbach-Linnik type problems on cubes of primes - The Ramanujan Journal, (2020). (The author uses our script but he does not cite our paper  !!)
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 the do not cite our paper  !!)
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.