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Beurling zeta functions, generalised primes, and fractal membranes

Hilberdink, T. W. and Lapidus, M. L. (2006) Beurling zeta functions, generalised primes, and fractal membranes. Acta Applicandae Mathematicae, 94 (1). pp. 21-48. ISSN 0167-8019

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We study generalised prime systems P (1 < p(1) <= p(2) <= ..., with p(j) is an element of R tending to infinity) and the associated Beurling zeta function zeta p(s) = Pi(infinity)(j=1)(1 - p(j)(-s))(-1). Under appropriate assumptions, we establish various analytic properties of zeta 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 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 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.

Item Type:Article
Divisions:Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:5075
Uncontrolled Keywords:Beurling (or generalised) primes and zeta functions Euler product analytic continuation functional equation Prime Number Theorem (with error term) partial orders on prime powers INVERSE SPECTRAL PROBLEMS WEYL-BERRY CONJECTURE DIRICHLET SERIES INTEGERS NUMBERS STRINGS DENSITY DRUMS

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