Sums of Higher Divisor Functions with Almost Equal Variables

Let k, ℓ ≥ 3 be integers, and let τ_k(n) denote the k-th divisor function. In this paper, we apply the circle method to obtain an asymptotic formula for the sum \(\sum_{|m_{i}-X|\leq Y}\ \ \tau_{k}(m_{1}^{2}+m_{2}^{2}+\cdots+m_{\ell}^{2})\ \ (i=1,2,\ldots,\ell)\) $$\sum _{|m_i-X|Y}{\tau }_k(m_1^2+m_2^2+\cdots +m_{\ell }^2)(i=1,2,\dots ,\ell )$$ with \(Y = {X^{1 - {8 \over {5(k + 1)}} + 20\varepsilon }}\) $$Y=X^{1-\frac{8}{5(k+1)}+20\epsilon }$$ tending to infinity. Previously, only the case k = 2 was considered.