About the Book
This dissertation, "Some Results of the Almost Goldbach Problems" by Chi-wai, Man, 文志偉, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author.
Abstract:
Abstract of thesis entitled 'SOME 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: Isthereapositiveintegerssuchthateverysufficientlylargeeveninteger N can be expressed as ν ν 1 k N =p +p +2 +---+2, (1) 1 2 wherep, p are primes andν, ---,ν are positive integers with k= 2. When g = 2, we obtain a better bound s
′ ′′ g. Moreover, let r (n) and r (n) be the numbers of representations of n in the form k k ν ν 1 k n=p+g +---+g (2) and ν ν
n=p +p +g +---+g (3) 1 2 respectively, where p, p, p are primes and ν, ---,ν are positive integers. 1 2 1 k iAbstract In Chapter 1 of this thesis, we obtain explicit formulas for computation of numerical boundsk of thek in (2) and (3) such that
(i) if k>=k andN>=N, then 0 k
k 2k X Y
D L D NL p-2
r (n)-
ϕ(D)logN ϕ(D) p-1 log N n2 wherethedashintheabovesumstandsforthetwoconditions(n, g)=(n-k, g-1)=1, ϕ(D) denotes the Euler-totient function, and (ii) if k>= 2k and (a) N>= N is even when g is even, or (b) N>= N satisfies N≡ k 0 k k (mod 2) when g is odd, then
NL ′′ r (N)>C,
log N where C is a positive constant depending on g only. For example, when 2g 2 3 4 5 6 7 8 9 10 k 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's 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.andWang, T.Z., Thenumberofpowersof2 ina 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.andWang, T.Z., Thenumberofpowersof2 ina representation of large even integers, II, Science in China, 41 (1998), 1255-1271. [Vi] Vinogradov, A. I., On an "almost binary" problem, Iz
