Volume 25 , Issue 1 , PP: 137-147, 2025 | Cite this article as | XML | Html | PDF | Full Length Article
Ahmed B. AL-Nafee 1 , Faez AL-Maamori 2
Doi: https://doi.org/10.54216/IJNS.250112
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.
Generalized Prime Systems , Beurling Zeta Function
[1] Beurling, A. (1937). Analyse de la loi asymptotique de la distribution des nombres premiers généralisés, I. Acta Math., 68, 255-291.
[2] Bateman, P. T., & Diamond, H. G. (2004). Analytic Number Theory. World Scientific Publishing.
[3] Diamond, H. G. (1970). Asymptotic distribution of Beurling's generalized integers. Illinois J. Math., 14, 12-28.
[4] Hilberdink, T. (2012). Generalised prime systems with periodic integer counting function. Acta Arith., 152, 217–241.
[5] Bateman, P. T., & Diamond, H. G. (1969). Asymptotic distribution of Beurling's generalized prime numbers. Studies in Number Theory, 6, 152–212.
[6] Diamond, H. G., & Zhang, W.B. (2012). A PNT equivalence for Beurling numbers. Functiones et Approximatio Commentarii Mathematici, 46(2), 225–234.
[7] Hall, R. S. (1967). Theorems about Beurling's generalized primes and the associated zeta function. PhD thesis, University of Illinois at Urbana-Champaign.
[8] Hilberdink, T. W. (2005). Well-behaved Beurling primes and integers. Journal of Number Theory, 112(2), 332–344.
[9] Zhang, W. B. (2007). Beurling primes with RH and Beurling primes with large oscillation. Mathematische Annalen, 337(3), 671–704.
[10] Al-Maamori, F., & Hilberdink, T. W. (2015). An example in Beurling’s theory of generalized primes. Acta Arith., 168(5), 383–396.
[11] Neamah, A. A., & Hilberdink, T. W. (2020). The average order of the Möbius function for Beurling primes. Int. J. Number Theory, 16(5), 1005–1011.
[12] Broucke, F., & Vindas, J. (Preprint, 2021). A new generalized prime random approximation procedure and some of its applications. Available on arXiv:2102.0847.
[13] Hilberdink, T. W., & Lapidus, M. L. (2006). Beurling zeta functions, Generalised Primes, and Fractal Membranes. Acta Appl Math, 94, 21-48.
[14] Smarandache, F. (1999). A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability. American Research Press: Rehoboth, NM.
[15] Smarandache, F. (2005). Neutrosophic set, a generalisation of the intuitionistic fuzzy sets. Int. J. Pure Appl. Math., 24, 287–297.
[16] Smarandache, F. (2013). Introduction to neutrosophic measure, neutrosophic integral, and neutrosophic probability. Infinite Study.
[17] Smarandache, F. (2020). "Neutrosophic Sets: Definitions, Properties, and Applications." Journal of Fuzzy Extension and Applications, Vol. 5, No. 1, pp. 22-31.
[18] Smarandache, F., & Vasantha Kandasamy, W. B. (2018). "Neutrosophic Algebraic Structures and Their Applications." Journal of Fuzzy Extension and Applications, Vol. 3, No. 2, pp. 104-117.
[19] AL-Nafee, A. B., Broumi, S., & Smarandache, F. (2021). "Neutrosophic Soft Bitopological Spaces." International Journal of Neutrosophic Science, Vol. 14, No. 1, pp. 47-56.
[20] AL-Nafee, A. B., Obeed, J. K., & Khalid, H. E. (2021). "Continuity and Compactness on Neutrosophic Soft Bitopological Spaces." International Journal of Neutrosophic Science, Vol. 16, No. 2, pp. 62-71.
