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Subsequence sums of zero-sum free sequences over finite abelian groups

Yongke Qu, Xingwu Xia, Lin Xue, Qinghai Zhong (2015)

Colloquium Mathematicae

Let G be a finite abelian group of rank r and let X be a zero-sum free sequence over G whose support supp(X) generates G. In 2009, Pixton proved that | Σ ( X ) | ≥ 2 r - 1 ( | X | - r + 2 ) - 1 for r ≤ 3. We show that this result also holds for abelian groups G of rank 4 if the smallest prime p dividing |G| satisfies p ≥ 13.

Sum of higher divisor function with prime summands

Yuchen Ding, Guang-Liang Zhou (2023)

Czechoslovak Mathematical Journal

Let l ⩾ 2 be an integer. Recently, Hu and Lü offered the asymptotic formula for the sum of the higher divisor function ∑ 1 ⩽ n 1 , n 2 , ... , n l ⩽ x 1 / 2 τ k ( n 1 2 + n 2 2 + ⋯ + n l 2 ) , where τ k ( n ) represents the k th divisor function. We give the Goldbach-type analogy of their result. That is to say, we investigate the asymptotic behavior of the sum ∑ 1 ⩽ p 1 , p 2 , ... , p l ⩽ x τ k ( p 1 + p 2 + ⋯ + p l ) , where p 1 , p 2 , ⋯ , p l are prime variables.

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