Goldbach Conjecture

The following sections are included:

• PREFACE

• CONTENTS

In a letter to Euler in 1742, Goldbach proposed two conjectures on the representations of integers as the sum of primes. These conjectures with some modifications may be stated as follows…

The following sections are included:

• Introduction.
  • The Farey dissection.

• Preliminary lemmas.

• Proof of the main theorems.
  • Approximation to v_r(n) by the singular series.
  • Summation of the singular series.
  • Proof of the final formulae.

• Remarks on ’Goldbach’s Theorem’.

Some simple examples of application of my method to the theory of primes were given in 1934 (¹)⋯

§1. In my note ’On the possibility of a method for some ‘additive’ and ‘distributive’ problems in the theory of prime numbers” [1], I sketch a proof of the Goldbach problem by the pure Riemann-Hadamard’s method of L-series and contour integration with some theorems on the density of zeros of L-series…

(A) By the circle method of Hardy and Littlewood and his method on the estimation of trigonometrical sum with prime variables, I. M. Vinogradov [l] first proved in 1937 that every large odd number is the sum of three prime numbers which is usually called the Goldbach-Vinogradov theorem or the three primes theorem. Later, Ju. V. Linnik [2] and N. G. Tchudakov [3] gave another two proofs on this theorem based on the estimation of the density of zeros of L-functions. Recently, H. L. Montgomery [4] and M. N. Huxley [5] gave two simplified proofs which are also based on the estimation of the density of zeros of L-functions, and in their proofs, the approximate functional equation of L-function and a mean value theorem on the fourth moment of L-function are used. In this paper, a new simplified analytic proof of the three primes theorem will be given which is not based on the Vinogradov’s estimation and the density theorem of zeros of L-function, and only some well-known simple facts on L-function are used…

The following sections are included:

• INTRODUCTION

• PROOFS OF THEOREMS 1 AND 2

• PROOFS OF THEOREMS 2 AND 4

• REFERENCES

§1. The theorem of Goldbach is well-known that one can write every even number as a sum of two prime numbers. In a letter of 1742, Euler has written: “I believe it is a completely acceptable theorem, although I cannot prove it.” This theorem has still not been proved, and it is the same about the following theorem: The sequence of the twin prime numbers is infinite. In an address delivered at the International Congress of Mathematics, Cambridge, 1912, E. Landau had said that he regarded these problems as “unattainable problems in modern science.”…

In 1919, V. Brun [l] gave a method for applying the sieve of Eratosthenes to a series of problems in number theory…

We use small Latin letters to denote the natural numbers which are also called numbers for simplicity, and p, q, s, t the prime numbers. Let

In the following we shall give a brief sketch of an elementary method which can be applied to the same problems as the “sieve method” which has been developed by BRUN. The new method is very simple and also generally yields better results than Bran’s…

For the sake of briefness, we write the following proposition by (a, b)…

The problem concerning the representations of even number as the sum of two primes and of odd number as the sum of three primes was suggested in the correspondence between Euler and Goldbach in 1742…

In this paper, we shall give the detailed proofs of certain results obtained upon assuming the truth of the grand Riemann hypothesis. (Cf. [1], [2]). First of all, let us state the grand Riemann hypothesis as follows:

(R) The real parts of all zeros of all Dirichlct’s L — functions L(s,χ) are ≤ 1/2.

§1. Let N be a large even number and V(m) the number of prime divisors of m. In 1948, Hungarian mathematician A. Rényi [1] has proved that N = a+b, where V(a) =1 and V(b)≤K in which K is an absolute constant. Under the assmuption of the Riemann hypothesis, Wang Yuan [2] has established that K≤3. We shall prove, in this paper, K≤5, namely…

Let π(x, D, ℓ) denote the number of primes in (1, x) which are ≡ ℓ(mod D). In this paper, a mean value theorem on π(x, D, ℓ) for (ℓ, D) =1 and “almost” all D ≤ x^{3/8 - ε} is proved, where ε is any pre-assigned positive number…

Goldbach-Euler problem concerning the representation of even number as the sum of two primes has not been solved up to date. With the aid of Eratosthenes sieve method and the theory of Dirichlet L-series developed by Ju. V. Linnik and his successors, one can prove that there exists an integer k such that eyery large number 2N can be represented as 2N = p+n, where p is a prime number and n has at most k prime factors. A. Rényi [1] first established the existence of such k. k = 4 was obtained by B. V. Levin, M. B. Barban, Wang Yuan and Pan Cheng Dong [2,3,4]. The corresponding results are also obtained on the problem of prime pairs, i.e., there exist infinitely many prime numbers p such that p+2 has at most k prime factors. In this paper, I shall prove k = 3…

§1. The basic aim of the paper is to prove the following theorem…

1. The purpose of this paper is to give a new and improved version of Linnik’s large sieve, with some applications. The large sieve has its roots in the Hardy-Littlewood method, and in its most general form it may be considered as an inequality which relates a singular series arising from an integral , where S(α) is any exponential sum, to the integral itself…

In this paper we shall prove that every sufficiently large even integer is a sum of a prime and a product of at most 2 primes. The method used is simple without any complicated numerical calculations.

The following sections are included:

• INTRODUCTION

• THE PROOF OF THEOREM 3

• APPLICATIONS

• REFERENCES

The following sections are included:

• References I

• References II