During the third decade of the last century, Arne Beurling stated the sense of the generalized prim systems. He said that any positive, increasingly real sequence started from any number greater than one precisely, and this sequence is called Beurling primes (or the generalized prim). They also mentioned that the fundamental Theorem of arithmetic gives the generalized integers. In the seventeenth decade of the last century, Diamond modified all the related counted functions of prime πg (x) and integers Ng (x) and the generalized Riemann zeta function. This work addresses an overview of the behavior of the generalized counting function Ng (x) where the change happens in the error term of Ng (x). The changing in E(x) = |Ng (x)-ρx| implies changing in the real part of (s = σ + it) and hence affects the behavior of the generalized ζg (x).