Blockchain and the Riemann Zeta Function

Abstract

Proof of Work (PoW) mechanisms used as part of the consensus mechanisms in block chains often have a major drawback namely that resources are only spent on doing the PoW and nothing else. In this paper we propose an adaption of hash based PoW’s. This adaption consists in two aspects, firstly by embedding the space of the hash’s, $\mathcal{H}$, in the space of complex numbers $\mathbb {C}$, more concretely in the subset $\mathcal{B}=\{z\in \mathbb {C}: 0<Re(z)<1\}$, $\Phi :\mathcal{H}\hookrightarrow \mathcal{B}$ and secondly in designing a new cryptographic puzzle in the hash space. The motivation behind these adaptions is that in this way the PoW can also be used to explore the Riemann Zeta Function (RZF). The RZF is associated with an open problem about the localization of its zeros. This problem was left as a conjecture by Riemann and is considered one of the most important open mathematical questions. The permanence of the problem is due to the mysterious behavior of the RZF in the region $\mathcal{B}$. This region will be by translation the search region used in the new cryptographic puzzle. The PoW will thus be able to contribute to a better understanding of the behaviour of the RZF in the region $\mathcal{B}$.