| مستخلص: |
Let x be a large real number, d (n) be the Dirichlet divisor function and Δ (x) = ∑ n ⩽ x d (n) − x log x − (2 γ − 1) x . In this paper, we consider a weighted form of the three primes theorem: S (N ; k 1 , k 2 , k 3) : = ∑ p 1 + p 2 + p 3 = N Δ k 1 (p 1) Δ k 2 (p 2) Δ k 3 (p 3) , where p 1 , p 2 , p 3 run over primes, k 1 , k 2 , k 3 ∈ { 1 , 2 , ⋯ , 9 } , and N is an large odd integer. For the case k i ∈ { 2 , 3 , ⋯ , 9 } (i = 1 , 2 , 3) , the others two k j ∈ { 2 , 3 , 4 } with j ≠ i , an asymptotic formula for S (N ; k 1 , k 2 , k 3) has been derived, along with a non-trivial upper bound estimate for S (N ; k 1 , k 2 , k 3) when min (k 1 , k 2 , k 3) = 1 . [ABSTRACT FROM AUTHOR] |
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