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Implementation of Pintz-Ruzsa method
for exponential sums over powers of two.
A. Languasco
and
A. Zaccagnini
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
-
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] and [5]. 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 - preprint, 2011, submitted.
[5] A. Rossi - The Goldbach-Linnik Problem: Some conditional results - PhD Thesis, Università of Milano 2011.
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.
Ultimo aggiornamento: 01.11.2011: 14:09:55
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