On certain trigonometric sums in several variables.

P. Haukkanen; R. Sivaramakrishnan

Displaying similar documents to “On certain trigonometric sums in several variables.”

The cyclotomic numbers of order eleven

Philip Leonard, Kenneth Williams (1975)

Acta Arithmetica

Similarity:

On the number of representations of an integer as a sum of primes belonging to given arithmetical progressions

A. Zulauf (1962-1964)

Compositio Mathematica

Similarity:

The ternary Goldbach problem in arithmetic progressions

Jianya Liu, Tao Zhan (1997)

Acta Arithmetica

Similarity:

For a large odd integer N and a positive integer r, define b = (b₁,b₂,b₃) and $(N, r) = b \in ℕ ³ : 1 \leq b_{j} \leq r, (b_{j}, r) = 1 a n d b ₁ + b ₂ + b ₃ \equiv N (m o d r) .$ It is known that $(N, r) = r ² \prod_{p | r} p | N ((p - 1) (p - 2) / p ²) \prod_{p | r} p ∤ N ((p ² - 3 p + 3) / p ²)$ . 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, ⎨ $p_{j} \equiv b_{j} (m o d r),$ j = 1,2,3,⎩ with prime variables is solvable whenever b ∈ (N,r), where A > 0 is arbitrary.