Displaying similar documents to “The divisor problem for arithmetic progressions with small modulus”

The ternary Goldbach problem.

David Rodney (Roger) Heath-Brown (1985)

Revista Matemática Iberoamericana

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The object of this paper is to present new proofs of the classical ternary theorems of additive prime number theory. Of these the best known is Vinogradov's result on the representation of odd numbers as the sums of three primes; other results will be discussed later. Earlier treatments of these problems used the Hardy-Littlewood circle method, and are highly analytical. In contrast, the method we use here is a (technically) elementary deduction from the Siegel-Walfisz Prime Number Theory....

The ternary Goldbach problem in arithmetic progressions

Jianya Liu, Tao Zhan (1997)

Acta Arithmetica

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For a large odd integer N and a positive integer r, define b = (b₁,b₂,b₃) and ( N , r ) = b ³ : 1 b j r , ( b j , r ) = 1 a n d b + b + b N ( m o d r ) . It is known that    ( N , r ) = r ² p | r p | N ( ( p - 1 ) ( p - 2 ) / p ² ) p | r p N ( ( p ² - 3 p + 3 ) / p ² ) . Let ε > 0 be arbitrary and R = N 1 / 8 - ε . We prove that for all positive integers r ≤ R, with at most O ( R l o g - A N ) exceptions, the Diophantine equation ⎧N = p₁+p₂+p₃, ⎨ p j b j ( m o d r ) , j = 1,2,3, ⎩ with prime variables is solvable whenever b ∈ (N,r), where A > 0 is arbitrary.