A Remark on the Riemann Hypothesis A.C.Manoharan [1] and Maithreyi. Manoharan [2] In profound memory of Arthur Strong Wightman [3] April 1, 2015 Abstract: We suggest a different way of examining the Riemann Hypothesis (RH). It concerns the BergmanShilov [4] boundary B of a domain D in several n complex variables theory, where n > 1. According to the RH the non-trivial zeros of the Riemann Zeta function ς(s) all lie on the critical line Re s = 1/2. We claim that the 1/2 which appears in the RH is intimately related to the 1/2 that appears in the dimension of B. In many situations dim(B) = (1/2) n , where "dim" refers to the dimension [5]. So in the case of the lowest possible n for several complex variables, n = 2, we have a topological space of 4 dimensions with dim(B) = (1/2) x 2 = 1, whereas the maximum topological boundary dimension in the general case is 3, for example in the case of a single hypersurface boundary. Often B(D) = B(H(D)) where H is the holomorphy envelope of D. So it is not even necessary [6] to calculate H(D) to determine B(D). The B boundary occurs in the case of a domain D bounded by analytic hypersurfaces, which is the case for Wightman Functions of orders 2 and 3 [7]. Then the hypersurface boundaries intersect one at a time, knocking down the dimension by 1 each time until the distinguished boundary of 1/2 the original dimension of D is reached. This is the approach for the Kallen-Toll dispersion relation representation [8]. Introduction: Over a long period of more than 150 years, efforts to solve the RH have generated enormous interest [9]. The truth of the RH is known as the greatest unsolved problem in mathematics. There is a solid connection between the RH and prime numbers and number theory [10]. Although the Riemann zeta function is a function of only one complex variable, it is possible to define related zeta functions over more variables [11], for example, generalizations of the Hurwitz zeta function. From our point of view of Holomorphy Envelopes of Domains, the Riemann Zeta function is only a special case (but of course the grandfather) of many zeta functions but those in turn are only special cases of all functions holomorphic in a domain D. Thus the Riemann Hypothesis is a special case of geometric analytic properties of Holomorphy Envelopes of Domains in Several Complex Variables [12]. As regards motivation from an intuitively naive point of view, we can say that the ½ in the Riemann Hypothesis is like saying the fact that one real dimension is ½ a complex dimension. References: 1)
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[email protected] and California State University, Stanislaus, Turlock CA 3) Notices of the American Mathematical Society, March 2015, p 249 4) Encyclopedia of Mathematics, Bergman-Shilov_boundary, Springer. 5) Dimension Theory, Hurewicz and Wallman, Princeton University Press (1941). 6) A.S.Wightman, Les Houches Lectures on Several Complex Variables, in Dispersion Relations and Elementary Particles, eds, de Witt and Omnes, Wiley (1960). 7) PCT Spin and Statistics and All That, R.F.Streater and A.S. Wightman, Benjamin (1963), and R.Jost, The General Theory of Quantized Fields, AMS (1965). 8) G.Kallen and J.S.Toll: Helv. Phys. Acta 33, 753, (1960). 9) In Search of the Riemann Zeros, M. L. Lapidus, AMS (2008). 10) T.M. Apostol: Introduction to Analytic Number Theory, Springer Verlag (1976), and Modular Functions and Dirichlet Series in Number Theory, 2nd Ed. Springer Verlag (1990). 11) Takashi Nakamura, arXiv:1405.1504v1.[math.NT], 7,May 2014. 12) For geometric properties of domains in several complex variables see: S.G. Krantz, Function Theory of Several Complex Variables 2nd edition AMS (2000).
