Article
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The Riemann Zeta Function of a Matrix/Tensor
Version 1
: Received: 30 September 2018 / Approved: 30 September 2018 / Online: 30 September 2018 (08:47:17 CEST)
How to cite: Sun, B. The Riemann Zeta Function of a Matrix/Tensor. Preprints 2018, 2018090604. https://doi.org/10.20944/preprints201809.0604.v1 Sun, B. The Riemann Zeta Function of a Matrix/Tensor. Preprints 2018, 2018090604. https://doi.org/10.20944/preprints201809.0604.v1
Abstract
This paper attempts to extend the Riemann Zeta function of a complex number to a function of a matrix and/or a tensor $A$, namely $$\zeta (A)=\sum _{n=1}^{\infty} \frac{1}{n^{A}}= \sum _{n=1}^{\infty} \sum_{k=1}^n\lambda_k A^k$$ and inverse $$A=\sum _{n=1}^{\infty} \sum_{k=1}^n \mu(n)\lambda_k \zeta(A^k)$$ where $\mu(n)$ is the M\"obius function, $A$ is a complex matrix or tensor with any order, and $\lambda_k$ is eigenvalue of the matri/tensor $A$. This kind of calculations on the Riemann Zeta function has never been seen in the literature. Some examples are provided.
Keywords
Riemann Zeta function; matrix; tensor
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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