On the general additive divisor problem

We obtain a new upper bound for the sum Σ h ≤ H Δ k ( N, h ) when 1 ≤ H ≤ N, k ∈ ℕ, k ≥ 3, where Δ k ( N, h ) is the (expected) error term in the asymptotic formula for Σ N < n ≤2 N d k ( n ) d k ( n + h ), and d k ( n ) is the divisor function generated by ζ ( s ) k . When k = 3, the result improves, for H ≥ N 1/2 , the bound given in a recent work of Baier, Browning, Marasingha and Zhao, who dealt with the case k = 3.