The average behaviour of the error term in a new kind of the divisor problem

We define a new kind of divisor function D(1)(n) by the nth coefficient of the Dirichlet series (ζ′(s))2 and denote by Δ(1)(x) the error term in the asymptotic formula for ∑n≤xD(1)(n). Then we compute the first moment of Δ(1)(x) that is ∫1XΔ(1)(x)dx and consider 'discrete mean values' ∑n≤xΔ(1)k(n) (k=1,2) and deduce asymptotic formulas which are analogous to the results obtained by Voronoï, Hardy and Furuya.