This work concerns the Goldbach-Linnik problem, which is a variation of Goldbach's one: here the goal is to prove that all large even integers can be written as a sum of two primes and k powers of 2, where k is a fixed positive integer. Assuming the Generalized Riemann Hypothesis, first we prove two results on the size of the exceptional set of the Goldbach-Linnik problem, which is the set of the even integer that can not be written as a sum of two primes and k powers of 2. To do this first we analyze the size of the exceptional set in short intervals, which are intervals of type [N;N+H] with N that tends to infinity and H = o(N), then in intervals of type [N; 2N]. Furthermore we study the Goldbach-Linnik problem from another point of view, that is that to find conditionally a value k0 such that every sufficiently large even integer has a representation as a sum of two primes and k0 powers of 2. Finally we analyze the real analogous of the Goldbach-Linnik problem, that concerns the numbers of the form l1*p1+l2*p2+t1*2^(m1)+...+ts*2^(ms) , with l1, l2, t1,...,ts real numbers not all equal to one and p1, p2 primes. Also in this case we study the problem conditionally. The main technique that we use to prove our results is the circle method applied in different way for every result. In fact, for example, we have to insert the method of Pintz-Ruzsa to do computations that involve the powers of 2 and to prove the last result we have to change the set of integration from S^1 to the real line, such that the method becomes different.
THE GOLDBACH-LINNIK PROBLEM: SOME CONDITIONAL RESULTS / A. Rossi ; tutor: Alessandro Zaccagnini; coordinatore: Marco Maria Peloso. Universita' degli Studi di Milano, 2011 Feb 11. 22. ciclo, Anno Accademico 2009. [10.13130/rossi-antonella_phd2011-02-11].
THE GOLDBACH-LINNIK PROBLEM: SOME CONDITIONAL RESULTS
A. Rossi
2011
Abstract
This work concerns the Goldbach-Linnik problem, which is a variation of Goldbach's one: here the goal is to prove that all large even integers can be written as a sum of two primes and k powers of 2, where k is a fixed positive integer. Assuming the Generalized Riemann Hypothesis, first we prove two results on the size of the exceptional set of the Goldbach-Linnik problem, which is the set of the even integer that can not be written as a sum of two primes and k powers of 2. To do this first we analyze the size of the exceptional set in short intervals, which are intervals of type [N;N+H] with N that tends to infinity and H = o(N), then in intervals of type [N; 2N]. Furthermore we study the Goldbach-Linnik problem from another point of view, that is that to find conditionally a value k0 such that every sufficiently large even integer has a representation as a sum of two primes and k0 powers of 2. Finally we analyze the real analogous of the Goldbach-Linnik problem, that concerns the numbers of the form l1*p1+l2*p2+t1*2^(m1)+...+ts*2^(ms) , with l1, l2, t1,...,ts real numbers not all equal to one and p1, p2 primes. Also in this case we study the problem conditionally. The main technique that we use to prove our results is the circle method applied in different way for every result. In fact, for example, we have to insert the method of Pintz-Ruzsa to do computations that involve the powers of 2 and to prove the last result we have to change the set of integration from S^1 to the real line, such that the method becomes different.File | Dimensione | Formato | |
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