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 (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 .