Opuscula Math. 37, no. 2 (2017), 225-263
http://dx.doi.org/10.7494/OpMath.2017.37.2.225
Opuscula Mathematica
Non-factorizable C-valued functions induced by finite connected graphs
Abstract. In this paper, we study factorizability of \(\mathbb{C}\)-valued formal series at fixed vertices, called the graph zeta functions, induced by the reduced length on the graph groupoids of given finite connected directed graphs. The construction of such functions is motivated by that of Redei zeta functions. In particular, we are interested in (i) "non-factorizability" of such functions, and (ii) certain factorizable functions induced by non-factorizable functions. By constructing factorizable functions from our non-factorizable functions, we study relations between graph zeta functions and well-known number-theoretic objects, the Riemann zeta function and the Euler totient function.
Keywords: directed graphs, graph groupoids, Redei zeta functions, graph zeta functions, non-factorizable graphs, gluing on graphs.
Mathematics Subject Classification: 05E15, 11G15, 11R47, 11R56, 46L10, 46L40, 46L54.
- D. Bump, Automorphic Forms and Representations, Cambridge Studies in Adv. Math. 55, Cambridge Univ. Press, 1996.
- I. Cho, Operators induced by prime numbers, Methods Appl. Math. Sci. 19 (2013) 4, 313-340.
- I. Cho, Algebras, Graphs and Their Applications, CRC Press, 2014.
- I. Cho, \(p\)-adic Banach space operators and adelic Banach space operators, Opuscula Math. 34 (2014) 1, 29-65.
- I. Cho, Classification on arithmetic functions and corresponding free-moment \(L\)-functions, Bulletin Korea Math. Soc. 52 (2015) 3, 717-734.
- I. Cho, \(\mathcal{C}\)-valued functions induced by graphs, Complex Anal. Oper. Theory 9 (2015) 3, 519-565.
- I. Cho, \(\mathcal{C}\)-valued functions induced by graphs and their factorization, Complex Anal. Oper. Theory (2015), DOI: 10.1007/s11785-015-0470-y. http://dx.doi.org/10.1007/s11785-015-0470-y
- I. Cho, P.E.T. Jorgensen, Moment computations of graphs with fractal property, J. Appl. Math. Comput. 37 (2011), 377-406.
- I. Cho, P.E.T. Jorgensen, Operators induced by graphs, Letters in Math. Phy. (2012), DOI: 10.1007/s11005-012-0575-4. http://dx.doi.org/10.1007/s11005-012-0575-4
- T. Gillespie, Prime number theorems for Rankin-Selberg \(L\)-functions over number fields, Sci. China Math. 54 (2011) 1, 35-46.
- J.P.S. Kung, M.R. Murty, G.-C. Rota, On the Rédei Zeta Function, J. Number Theor. 12 (1980), 421-436.
- V.S. Vladimirov, I.V. Volovich, E.I. Zelenov, \(p\)-Adic Analysis and Mathematical Physics, Ser. Soviet & East European Math., vol. 1, World Scientific, 1994.
- D.V. Voiculescu, K.J. Dykema, A. Nica, Free Random Variables, CRM Monograph Series, vol. 1, Amer. Math. Soc., 2002.
- Ilwoo Cho
- St. Ambrose University, Department of Mathematics and Statistics, 421 Ambrose Hall, 518 W. Locust St., Davenport, Iowa, 52803, USA
- Communicated by P.A. Cojuhari.
- Received: 2016-03-07.
- Revised: 2016-07-21.
- Accepted: 2016-08-13.
- Published online: 2017-01-03.
