Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed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. 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

Comment 1
Commenter: Peter P
The commenter has declared there is no conflict of interests.
Comment: Hello, I had been looking into extending the definition of the dirichlet eta function to matrices/tensor as well. I used diagonalization and the jordan form (if needed) of the matrices to find e^A. Using 2x2 matrices that are isomorphic to complex numbers, I was able to obtain the same results as I would by only using complex numbers in the dirichlet eta func. I found results for 3x3 matrices as well, but I’m unsure if my methods are justified, mathematically speaking. Let me know your thoughts!
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