Some results of the almost Goldbach problems

by Man, Chi-wai

Abstract (Summary)
(Uncorrected OCR) Abstract of thesis entitled ?OME RESULTS ON THE ALMOST GOLDBACH PROBLEMS?submitted by Man Chi Wai for the degree of Master of Philosophy at the University of Hong Kong in February, 2000. The Almost Goldbach Problem was first considered by Yu. V. Linnik in 1951. He dealt with the following problem: Is there a positive integer s such that every sufficiently large even integer N can be expressed as N = p1+p2+2Vl+--- + 2Vk, (1) where p1,p2 are primes and ?1, ???, ?k are positive integers with k < s? The existence of s (that is the qualitative part of the Almost Goldbach Problem) was settled by Linnik [L1 & L2]. The results of Linnik were generalized by A. I. Vinogradov [Vi] in 1956 and by Gallagher [G2] in 1975. The problem on the numerical value of the upper bound for s (i.e. the quantitative part of the Almost Goldbach Problem) was recently considered by J. Y. Liu, M. C. Liu and T. Z. Wang. They obtained a numerical bound for s under the Generalized Riemann Hypothesis (GRH) in [LLW1] and without the GRH in [LLW2]. In this thesis, we always assume the GRH. We shall use the circle method refined by Gallagher [G2] and a well-known result of J. R. Chen [C] obtained by sieve methods in the Goldbach problem to generalize the results in [LLW1]. We can replace the 2 in (1) by any integer g > 2. When g = 2, we obtain a better bound s < 722 while in [LLW1] they obtained s < 770. Let D = g(g-1),L = [logfl N] and Nk be some positive integer depending on k and g. Moreover, let r'k(n) and r?n) be the numbers of representations of n in the form n = p + gVl +---+gVk (2) and n = p1+p2+gVl +---+gUk (3) respectively, where p,p1,p2 are primes and ?1, ???, ?k are positive integers. i Abstract In Chapter 1 of this thesis, we obtain explicit formulas for computation of numerical bounds k0 of the k in (2) and (3) such that (i) if k>k0 and N > Nk, then D Lk \2 D NL2k (p-2 n2 , where the dash in the above sum stands for the two conditions (n, g) = (n-k,g-1) = 1, ?(D) denotes the Euler-totient function, and (ii) if k > 2k0 and (a) N > Nk is even when g is even, or (b) N > Nk satisfies N = k (mod 2) when g is odd, then i(N) > C NLk log2 N , where C is a positive constant depending on g only. For example, when 2 < g < 10, we have the following k0: g 2 3 4 5 6 7 8 9 10 k0 361 3287 5413 21893 20791 239273 50772 294722 211836 For the sake of easy reference by the readers, we will give a detailed proof of [G2, Theorem 3] in Chapter 2 of this thesis. References: [C] Chen, J. R., On Goldbach? problem and the sieve methods, Sci. Sin., 21 (1978), 701-739. [G2] Gallagher, P. X., Primes and powers of 2, Invent. Math. 29 (1975), 125-142. [L1] Linnik, Yu. V., Prime numbers and powers of two, Trudy Mat. Inst. Steklov, 38(1951), 151-169. [L2] Linnik, Yu. V., Addition of prime numbers and powers of one and the same number, Mat. Sb. (N. S.), 32(1953), 3-60. [LLW1] Liu, J. Y., Liu, M. C. and Wang, T. Z., The number of powers of 2 in a representation of large even integers, I, Science in China, Series A, 41 (1998), 386-398; (Chinese version) 28 (1998), 1-13. [LLW2] Liu, J. Y., Liu, M. C. and Wang, T. Z., The number of powers of 2 in a representation of large even integers, II, Science in China, 41 (1998), 1255-1271. [Vi] Vinogradov, A. I., On an ?lmost binary?problem, Izv. Akad. Nauk. SSSR, Ser. Mat., 20(1956), 713-750. ii
Bibliographical Information:


School:The University of Hong Kong

School Location:China - Hong Kong SAR

Source Type:Master's Thesis

Keywords:number theory


Date of Publication:01/01/2000

© 2009 All Rights Reserved.