Beurling Zeta Functions, Generalised Primes, and Fractal Membranes

Abstract

We study generalised prime systems $\mathcal{P}$ $(1<p_1\le p_2\le ...,$ with $p_j\in \mathbb{R}$ tending to infinity) and the associated Beurling zeta function ${\zeta }_{\mathcal{P}}(s)={\prod }_{j=1}^{\infty }(1-p_j^{-s}{)}^{-1}$. Under appropriate assumptions, we establish various analytic properties of ${\zeta }_{\mathcal{P}}(s)$, including its analytic continuation and we characterise the existence of a suitable generalised functional equation. In particular, we examine the relationship between a counterpart of the Prime Number Theorem (with error term) and the properties of the analytic continuation of ${\zeta }_{\mathcal{P}}(s)$. Further we study `well-behaved' g-prime systems, namely, systems for which both the prime and integer counting function are asymptotically well-behaved. Finally, we show that there exists a natural correspondence between generalised prime systems and suitable orders on ${\mathbb{N}}^2$. Some of the above results may be relevant to the second author's theory of `fractal membranes', whose spectral partition functions are precisely given by Beurling zeta functions.