Abstract
For J 3 ( n ) = ∑ p 1 + p 2 + p 3 = n p 1 ≡ a 1 ( mod q 1 ) log p 1 log p 2 log p 3 , it is shown that ∑ q 1 ⩽ n θ max ( a 1 , q 1 ) = 1 J 3 ( n ) - n 2 2 ϕ ( q 1 ) S 3 ( n ) ≪ A n 2 ( log n ) - A for any A and any θ < 1 / 2 , what improves a work of Tolev; S 3 ( n ) is the corresponding singular series. A special form of a sieve of Montgomery is used.
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