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Quantum physics and number theory connected by the Riemann zeta function

C. Feiler

Published Jan 1, 2013• 4 citations

Abstract

The intimate connection between the zeros of the Riemann zeta function and the distribution of prime numbers makes this function famous beyond the borders of mathematics. Since the precise location of the non-trivial zeros (Riemann hypothesis) is still unconfirmed many efforts are directed to the investigation of the zeta function, not only with mathematical but also with physical methods. The physical approach in this thesis is inspired by the interaction of a cavity field with a two-level atom described by the Jaynes-Cummings-Paul model. However, in our case the interaction Hamiltonian depends logarithmically on the photon number operator. The time-evolved quantum states produce with appropriately chosen reference states the values of the Riemann zeta function. It is shown that in the complex half-plane with real parts larger than one, the cavity field suffices to create the zeta function whereas the analytical continuation of the zeta function can only be realized with entangled states. Moreover, the continuous Newton method is used to illustrate the behavior of the Riemann zeta function in the complex plane. This method depicts a complex function F by lines of constant phase which start at the poles of the function and lead into its zeros. In the case of the Riemann zeta function, this method would reveal if the Riemann hypothesis was violated.

DOI: 10.18725/oparu-2645

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The Riemann Hypothesis, a long-standing unsolved problem in mathematics, concerns the distribution of prime numbers. Despite extensive research, no definitive proof remains elusive. This paper explores a novel quantum algorithm designed to approximate the Riemann zeta function, aiming to provide exponential speedups over classical methods. We propose a hybrid quantum-classical approach, utilizing quantum superposition and entanglement to efficiently explore the zeta function's complex landscape, ultimately leading to a significant computational advantage. This research focuses on the fundamental potential of quantum computation to address this mathematical challenge.

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