International Journal of Neutrosophic Science IJNS 2690-6805 2692-6148 10.54216/IJNS https://www.americaspg.com/journals/show/2939 2020 2020 Ministry of Education Open Educational College, Department of Mathematics, Babylon, Iraq Ahmed Ahmed Department of computer science,Babylon University, Babylon, Iraq Faez AL AL-Maamori Beurling (or generalized) prime system has been defined by Arne Beurling in 1937, and several couthers have been working on this during the last century. This work focuses on addressing some concrete examples of an outer generalized prime system involving Beurling zeta function. The core of this work is to create a discrete generalized prime system under a fixed condition to give a new upper bound for Beurling zeta function. 2025 2025 137 147 10.54216/IJNS.250112 https://www.americaspg.com/articleinfo/21/show/2939