aspg-21-2939-1791475890 20261008161130 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group International Journal of Neutrosophic Science IJNS 2692-6148 2690-6805 2025 25 1 Ahmed B. AL-Nafee Ministry of Education Open Educational College, Department of Mathematics, Babylon, Iraq Faez AL-Maamori Department of computer science,Babylon University, Babylon, Iraq 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 137 147 2939 https://creativecommons.org/licenses/by/4.0/ 10.54216/IJNS.250112 https://www.americaspg.com/journal/21/article/2939 https://www.americaspg.com/storage/01727303644.pdf