Beurling Zeta Functions, Generalised Primes, and Fractal Membranes

We study generalised prime systems P (1 < p1 ≤ p2 ≤ · · · , with pj ∈ R tending to infinity) and the associated Beurling zeta function ζP(s) = ∏∞ j=1(1 − p−s j )−1. Under appropriate assumptions, we establish various analytic properties of ζP(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 ζP(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 N2. 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.