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We refine a previous work of K. Matsumoto and H. Ishikawa, obtaining anasymptotic formula for the mean square of the product of the Riemannzeta-function and a Dirichlet polynomial in the critical strip(1/4<$\sigma$<1/2), by obtaining an explicit formula of Atkinson type for itserror term. This work is closely related to the generalized Dirichlet divisorproblem. The form of these formulas is akin to, yet more complex than,Voronoi's formulas in the divisor problem.

The mean square of the product of the Riemann zeta-function and a Dirichlet polynomial in the critical strip