We follow a paper by Sedunova regarding Vaughan’s basic mean value Theorem to improve and complete more general demonstration for suitable class of arithmetic functions as started Cojocaru Murty. As an application we derive theorem the von Mangoldt generalized functions.

We consider   bivariate  mean-value interpolationproblem, where the integrals over circles are interpolation data. In this case the problem  is described  over   circles of the same radius and with  centers are on astraight line and it is shown that in this case the interpolation is not correct.

we consider   bivariate  mean-value interpolationproblem, where the integrals over circles are interpolation data. in this case the problem  is described  over   circles of the same radius and with  centers are on astraight line and it is shown that in this case the interpolation is not correct.