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In 1956, Tong established an asymptotic formula for the mean square of theerror term in the summatory function of the Piltz divisor function $d_3(n).$The aim of this paper is to generalize Tong's method to a class of Dirichletseries that satisfy a functional equation. As an application, we can establishthe asymptotic formulas for the mean square of the error terms for a class offunctions in the well-known Selberg class. The Tong-type identity and formulaestablished in this paper can be viewed as an analogue of the well-knownVorono\"i's formula.

On the Tong-type identity and the mean square of the error term for an extended Selberg class