(untitled) (JOURNAL OF UNIVERSITY OF BABYLON for Pure and Applied Sciences, 2017)
In this article we focus on the behaviors of the generalised counting function of primes (x) and the counting function of integers (x) as well as the link between them as x . Here the Riemann zeta function (s) ( = , (s) > 1 ) play an important role as a link between (x) and (x) . This work will go through the method ( not in details ) adapted by Balanzario [Balanzario , 1998] and later generalised by AL- Maamori [AL- Maamori , 2013 ] . Finally we shall draw a diagram in order to determine the relation between and , (where and are the power of the error terms H1(x) , H2(x) of (x) and (x) respectively) . The aim of this work is to analysis the behaviour of (x) and (x) as x . Note that : ʺ It’s a beneficial to point out that our effort in this paper is not to exchange the values of some functions of Balanzarioʹs method . Since , changing any small value of one of the functions of Balanzarioʹs method may be leads to loss the aim of the work ʺ . Therefore , in this article we show the ability of changing the values of some functions and in which places in the proof we should sort out .