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Yasuhiro Abe, Ph.D.

Cereja Technology Co., Ltd.
3-11-15 UEDA-Bldg. 4F, Iidabashi
Chiyoda-ku, Tokyo 102-0072, Japan

Physical interpretation of the Riemann hypothesis

The Riemann hypothesis is an important unsolved problem in mathematics. As well-known, the problem itself is quite simple: "To show the real part of any nontrivial zeros of Riemann's zeta function is equal to 1/2". But it has been unsolved for over 150 years. Of particular importance of this problem is that it is closely related to the distribution of prime numbers. One of the indications of this relation is that the zeta function can be expressed in terms of a product sum over prime numbers via the formula of Euler's product. In what follows, we would like to introduce how we can interpret the Riemann hypothesis in the framework of holonomy formalism by use of a correspondence between primes and knots.

The holonomy formalism is a kind of universal formalism that uses a holonomy operator of conformal field theory as a generating functional for scattering amplitudes of massless bosons. (For its background and formulation, see here.) Thus, in applying this formalism to the Riemann hypothesis, we first need to define "prime-creation" operators. For this purpose, it is necessary to develop quantum-theoretical treatments for various quantities —— such as Legendre symbols, Jacobi symbols and Gauss sums —— that appear in the elementary theory of numbers.

On the other hand, it is known that Gauss's linking number appearing in Maxwell's theory of electromagnetism can easily be obtained in terms of abelian Chern-Simons theory that is closely related to conformal field theory. Since we can consider the holonomy operator as a generalization of Chern-Simons action, this fact suggests that we can obtain a generalization of linking numbers in terms of abelian holonomy operators. Indeed, if we identify the Riemann surface on which the two-dimensional conformal field theory is defined as a torus and consider holonomy operators of its zero modes, then we can express a generalized linking number in terms of such zero-mode holonomy operators.

Applying a mathematically-known correspondence between primes and knots or, more precisely, a correspondence between linking numbers and Legendre symbols, we can then incorporate the Legendre symbols as operators into the zero-mode holonomy operators. This can be thought of as an application of abelian holonomy formalism to elementary number theory. Furthermore, since in number theory it is known that the Gauss sum can be considered as a Fourier transform of the Legendre symbol, we can interpret the Gauss sum as an operator that is canonically conjugate to the Legendre symbol. The Gauss sum is a quantity that depends only on an odd prime and its modulus gives the square root of the odd prime. In the holonomy formalism under consideration, information on odd primes is encoded only in the Gauss-sum operator (or, more precisely, certain powers of it). Therefore it is physically natural to interpret this operator as an operator that is relevant to creation of primes.

From these arguments we find that one can define a "prime-creation" operator but how it connects with the Riemann hypothesis has not been mentioned at all. However, in actuality, an explicit calculation of the zero-mode holonomy operator shows that it can be expressed in terms of an iterated-integral representation of Riemann's zeta function. Thus, using the formula of Euler's product and replacing its prime-number part by the Gauss-sum operators, we can obtain a quantum realization of the zeta function in the abelian holonomy formalism. Furthermore, by use of this new zero-mode holonomy operator, we can calculate "scattering amplitudes" for identical odd primes within the framework of holonomy formalism. When there is only one prime to be scattered, the "scattering amplitude" of course vanishes. The Riemann hypothesis can be interpreted by means of this physically obvious fact, i.e., there is no notion of "scattering" for a single-particle system. For details of these formulations, please see arXiv:1005.4299.