Preprint Article Version 1 This version is not peer-reviewed

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 (doi: 10.20944/preprints201809.0604.v1). Sun, B. The Riemann Zeta Function of a Matrix/Tensor. Preprints 2018, 2018090604 (doi: 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.

Subject Areas

Riemann Zeta function; matrix; tensor

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