# On primeness of the Selberg zeta-function

## TL;DR

This paper proves that the Selberg zeta-function for a compact Riemann surface exhibits pseudo-primeness and right-primeness, revealing new algebraic properties of this important function in spectral geometry.

## Contribution

It establishes the pseudo-prime and right-prime nature of the Selberg zeta-function, a novel algebraic insight into its structure.

## Key findings

- Selberg zeta-function is pseudo-prime.
- Selberg zeta-function is right-prime.
- Provides new understanding of the algebraic structure of the zeta-function.

## Abstract

In this note we prove that the Selberg zeta-function associated to a compact Riemann surface is pseudo-prime and right-prime in the sense of a decomposition.