The Behaviors of some Counting Functions of g-primes and g-integers as x goes to Infinity

Abstract

In this article we focus on the behaviors of the generalised counting function of primes 〖 Π〗_P(x) and the counting function of integers N_P(x) as well as the link between them as x ⟶ ∞ . Here the Riemann zeta function ζ_P(s) ( = ∑_n▒n^(-s) , R(s) > 1 ) play an important role as a link between 〖 Π〗_P(x) and N_P(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 α and β , (where α and β are the power of the error terms H1(x) , H2(x) of 〖 Π〗_P(x) and N_P(x) respectively) . The aim of this work is to analysis the behaviour of 〖 Π〗_P(x) and N_P(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 .