Displaying similar documents to “The divisor function d 3 ( n ) in arithmetic progressions”

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 ∈ ℕ ³ : 1 ≤ b j ≤ r , ( b j , r ) = 1 a n d b ₁ + b ₂ + b ₃ ≡ 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.

On normal lattice configurations and simultaneously normal numbers

Mordechay B. Levin (2001)

Journal de théorie des nombres de Bordeaux

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Let q , q 1 , ⋯ , q s ≥ 2 be integers, and let α 1 , α 2 , ⋯ be a sequence of real numbers. In this paper we prove that the lower bound of the discrepancy of the double sequence ( α m q n , ⋯ , α m + s - 1 q n ) m , n = 1 M N coincides (up to a logarithmic factor) with the lower bound of the discrepancy of ordinary sequences ( x n ) n = 1 M N in s -dimensional unit cube ( s , M , N = 1 , 2 , ⋯ ) . We also find a lower bound of the discrepancy (up to a logarithmic factor) of the sequence ( α 1 q 1 n , ⋯ , α s q s n ) n = 1 N (Korobov’s problem).

A short intervals result for linear equations in two prime variables.

M. B. S. Laporta (1997)

Revista Matemática de la Universidad Complutense de Madrid

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Given A and B integers relatively prime, we prove that almost all integers n in an interval of the form [N, N+H], where N exp(1/3+e) ≤ H ≤ N can be written as a sum Ap1 + Bp2 = n, with p1 and p2 primes and e an arbitrary positive constant. This generalizes the results of Perelli et al. (1985) established in the classical case A=B=1 (Goldbach's problem).