The behaviors of some counting functions of g-primes and g-integers as x goes to infinity

Abstract EN

-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 Bal- anzario [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 Balanzarioʹ s method .

Since , changing any small value of one of the functions of Balanzarioʹ s 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 .