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Keywords:
higher divisor function; circle method; prime
higher divisor function; circle method; prime
Summary:
Let $l\geqslant 2$ be an integer. Recently, Hu and Lü offered the asymptotic formula for the sum of the higher divisor function $$ \sum _{1\leqslant n_{1},n_{2},\ldots ,n_{l}\leqslant x^{1/2}}\tau _{k}(n_{1}^{2}+n_{2}^{2}+\cdots +n_{l}^{2}), $$ where $\tau _{k}(n)$ 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 $$ \sum _{1\leqslant p_{1},p_{2},\ldots ,p_{l}\leqslant x}\tau _{k}(p_{1}+p_{2}+\cdots +p_{l}), $$ where $p_1,p_2,\dots ,p_l$ are prime variables.
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