Research ArticleOpen AccessGoogle Scholar indexed
Euler Product Expressions of Absolute Tensor Products of Dirichlet L -Functions
Department of Biomedical Engineering, Toyo University, Saitama, Japan
Department of Mechanical Engineering, Toyo University, Saitama, Japan
- 1 Department of Biomedical Engineering, Toyo University, Saitama, Japan
- 2 Department of Mechanical Engineering, Toyo University, Saitama, Japan
Advances in Pure Mathematics·Volume 14 (2024)·Pages 451–486·Published 4 June 2024·DOI10.4236/apm.2024.146026
Copy link · social · email
Abstract
In this paper, we calculate the absolute tensor square of the Dirichlet L -functions and show that it is expressed as an Euler product over pairs of primes. The method is to construct an equation to link primes to a series which has the factors of the absolute tensor product of the Dirichlet L -functions. This study is a generalization of Akatsuka’s theorem on the Riemann zeta function, and gives a proof of Kurokawa’s prediction proposed in 1992.
KeywordsDirichlet L -FunctionAbsolute Tensor Product (Kurokawa Tensor Product)Euler Product
- Kurokawa, N. (1992) Multiple Zeta Functions: An Example. Advanced Studies in Pure Mathematics , 21, 219-226. https://doi.org/10.2969/aspm/02110219
- Deninger, C. (1992) Local L-Factors of Motives and Regularized Determinants. I n ventiones Mathematicae , 107, 135-150. https://doi.org/10.1007/bf01231885
- Kurokawa, N. and Wakayama, M. (2004) Zeta Regularizations. Acta Applicandae Mathematicae , 81, 147-166. https://doi.org/10.1023/b:acap.0000024207.37694.3b
- Koyama, S. and Kurokawa, N. (2006) Multiple Euler Products. American Math e matical Society Translations : Series 2, 218, 101-140.
- Akatsuka, H. (2005) Euler Product Expression of Triple Zeta Functions. Intern a tional Journal of Mathematics , 16, 111-136. https://doi.org/10.1142/s0129167x05002825
- Kurokawa, N. and Wakayama, M. (2004) Absolute Tensor Products. International Mathematics Research Notices , 2004, 249-260. https://doi.org/10.1155/s1073792804132327
- Akatsuka, H. (2009) The Double Riemann Zeta Function. Communications in Number Theory and Physics , 3, 619-653. https://doi.org/10.4310/cntp.2009.v3.n4.a2.
- Cramér, H. (1919) Studien über die Nulstellen der Riemannschen Zetafunktion. Mathematische Zeitschrift , 4, 104-130.
- Jameson, G.J.O. (2003). The Prime Number Theorem. Cambridge University Press, Cambridge. https://doi.org/10.1017/cbo9781139164986
- M. Riesz, M. (1916) Ein konvergenzsatz für Dirichletsche Reihen. Acta Mathemat i ca , 40, 349-361. https://doi.org/10.1007/bf02418551
- Guinand, A.P. (1949) Fourier Reciprocities and the Riemann Zeta-Function. Pr o ceedings of the London Mathematical Society , 2, 401-414. https://doi.org/10.1112/plms/s2-51.6.401
- Kaczorowski, J. (1990) The K-Functions in Multiplicative Number Theory. I. on Complex Explicit Formulae. Acta Arithmetica , 56, 195-211. https://doi.org/10.4064/aa-56-3-195-211
- Montgomery, H.L. and Vaughan, R.C. (2006) Multiplicative Number Theory I. Cambridge University Press, Cambridge. https://doi.org/10.1017/cbo9780511618314
- Hirano, M., Kurokawa, N. and Wakayama, M. (2003) Half Zeta Functions. Journal of the Ramanujan Mathematical Society , 18, 195-209.