Beurling Zeta Functions, Generalised Primes, and Fractal Membranes

We study generalised prime systems $\mathcal{P}$ $(1<p_1\leq p_2\leq...,$ with $p_j\in\R$ 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 $\N^2$. 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.