According to Beurling in 1937, a generalized prime is any increasing infinite positive real sequence for which the first element is precisely greater than 1. Additionally, the basic fundamental theorem of arithmetic can be used to construct the sequence of Beurling integers. The example of a discrete and continuing Beurling's prime system is the main topic of this article. The challenge in this approach is finding a discrete system from a continues prime system. This work demonstrates the connection between Beurling Zeta function and Beurling counting functions which discusses the error -term of N(x) and π(x) and the relation between them with an estimation in order to obtain a better error-term n comparison with what are known till now.

(Analytic Number Theory, generalized prime systems