Tokyo Journal of Mathematics

A Bicomplex Riemann Zeta Function

Dominic ROCHON

Full-text: Open access


In this work we use a commutative generalization of complex numbers, called bicomplex numbers, to introduce a holomorphic Riemann zeta function of two complex variables satisfying the complexified Cauchy-Riemann equations. Furthermore, we establish a bicomplex Riemann hypothesis equivalent to the complex Riemann hypothesis of one variable and we obtain a bicomplex Euler Product.

Article information

Tokyo J. of Math. Volume 27, Number 2 (2004), 357-369.

First available in Project Euclid: 5 June 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30G35: Functions of hypercomplex variables and generalized variables
Secondary: 32A30: Other generalizations of function theory of one complex variable (should also be assigned at least one classification number from Section 30) {For functions of several hypercomplex variables, see 30G35}


ROCHON, Dominic. A Bicomplex Riemann Zeta Function. Tokyo J. of Math. 27 (2004), no. 2, 357--369. doi:10.3836/tjm/1244208394.

Export citation


  • L. V. Ahlfors, Complex Analysis, 3$^{e}$ ed, McGraw-Hill (1979).
  • R. Fueter, Analytische Funktionen einer Quaternionen variablen, Comment Math. Helv. 4 (1932), 9–20.
  • I. L. Kantor, Hypercomplex numbers, Springer (1989).
  • G. Moisil, Sur les quaternions monogenes, Bull. Sci. Math. Paris 55 No. 2 (1931), 169–194.
  • G. Moisil and N. Theodoresco, Fonctions holomorphes dans l'espace, Mathematica $($Cluj$)$ 5 (1931), 142–159.
  • S. J. Patterson, An Introduction to the theory of the Riemann Zeta-Function, Cambridge University Press (1988).
  • G. B. Price, An Introduction to Multicomplex Spaces and Functions, Marcel Dekker (1991).
  • R. M. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, Springer (1986).
  • D. Rochon, A Bloch Constant for Hyperholomorphic Functions, Complex Variables 44 (2001), 85–101.
  • D. Rochon, Sur une généralisation des nombres complexes: les tétranombres, M. Sc. Université de Montréal, (1997).
  • J. Ryan, Complexified Clifford Analysis, Complex Variables 1 (1982), 119–149.
  • M. V. Shapiro and N. L. Vasilevski, Quaternionic $\psi$-hyperholomorphic functions, singular integral operators and boundary value problems. I. $\psi$-hyperholomorphic function theory, Complex Variables 27 (1995), 17–46.
  • M. V. Shapiro and N. L. Vasilevski, Quaternionic $\psi$-hyperholomorphic functions, singular integral operators and boundary value problems. II. Algebras of singular integral operators and Riemann type boundary value problems, Complex Variables 27 (1995), 67–96.
  • C. Segre, Le Rappresentazioni Reali delle Forme Complesse a Gli Enti Iperalgebrici, Math. Ann. 40 (1892), 413–467.
  • B. V. Shabat, Introduction to Complex Analysis part II: Functions of Several Variables, American Mathematical Society (1992).
  • H. Shimizu, On Zeta Function of Quaternion Algebras, The Annals of Mathematics 81 (1965), 166–193.
  • G. Sobczyk, The Hyperbolic Number Plane, The College Mathematics Journal 26 (1995), 268–280.