Beurling Zeta Functions, Generalised Primes, and Fractal Membranes
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Beurling Zeta Functions, Generalised Primes, and Fractal Membranes Titus W. Hilberdink · Michel L. Lapidus
Received: 9 March 2005 / Accepted: 3 July 2006 / Published online: 27 September 2006 © Springer Science + Business Media B.V. 2006
Abstract 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 − −1 ) . Under appropriate assumptions, we establish various analytic properties of p−s j 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 are relevant to the second author’s theory of ‘fractal membranes’, whose spectral partition functions are given by Beurling-type zeta functions, as well as to joint work of that author and R. Nest on zeta functions attached to quasicrystals. Key words Beurling (or generalised) primes and zeta functions · Euler product · analytic continuation · functional equation · Prime Number Theorem (with error term) · partial orders on prime powers. Mathematics Subject Classifications (2000) Primary: 11M41 · 11N80 · Secondary: 11M06 · 11N05 · 11S45.
The work of M. L. Lapidus was partially supported by the U. S. National Science Foundation under grant DMS-0070497. T. W. Hilberdink (B) Department of Mathematics, University of Reading, Whiteknights, PO Box 220, Reading RG6 6AX, UK e-mail: [email protected] M. L. Lapidus Department of Mathematics, University of California, Riverside, CA 92521-0135, USA e-mail: [email protected]
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Acta Appl Math (2006) 94: 21–48
1 Introduction 1.1 Generalised Primes and Beurling Zeta Functions A generalised prime system P is a sequence of positive reals p1 , p2 , p3 , . . . satisfying 1 < p1 ≤ p2 ≤ · · · ≤ pn ≤ · · · and for which pn → ∞ as n → ∞. From these can be formed the system N of generalised integers or Beurling integers; that is, the numbers of the form pa11 pa22 . . . pakk where k ∈ N and a1 , . . . , ak ∈ N0 .1 For simplicity, we shall often just refer to g-primes and g-integers. This system generalises the notion of prime numbers and the natural numbers obtained from them. Such systems (along with the attached zeta functions) were first introduced by Beurling [4] and have been studied by numerous authors since then (see, in particular, the papers by Bateman and Diamond [1], Diamond [9–12], Hall [16, 17], Malliavin [36], Nyman [37] and Lagarias [22]). Recently, new connections have been proposed between suitable Beurling-type zeta functions and several subjects relevant to potential applications or a
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