## Beurling zeta functions, generalised primes, and fractal membranes.(English)Zbl 1133.11057

Let $$P=\{1<p_1\leq p_2\leq \cdots\leq p_k\leq p_{k+1}\to\infty\}$$ be a sequence of real numbers (generalized primes), and let $$N$$ be the set of numbers which are finite products of elements of $$P$$ (generalized or Beurling numbers). Furthermore, $$P(x)$$ and $$N(x)$$ denote the counting functions of the sets $$P$$ and $$N$$, respectively. There are a lot of investigations on the behavior of $$P(x)$$ under some assumptions about $$N(x)$$ for $$x\to\infty$$ and vice versa during the last 70 years, after A. Beurling has published his paper on the distribution of generalized primes.
In the note under review the authors consider the connection between analytic properties of the associated zeta functions such as analytic continuation or functional equation and the distribution of the generalized primes. Further, they study ‘well-behaved’ systems, i.e. systems for which $$\alpha,\beta\in[0,1)$$ exist with $\psi(x)=\sum_{p_j^a\leq x}\log p_j=x + O(x^{\alpha+\varepsilon})$ and $N(x)=\rho x+O(x^{\beta+\varepsilon})\quad (\text{for some }\;\rho>0)$ $$\varepsilon>0$$ arbitrary.
In a forthcoming book – entitled “In Search of the Riemann Zeros: Strings, Fractal Membranes, and Noncommutative Spacetimes” Providence, RI: AMS (2008; Zbl 1150.11003) – by the second author it is shown (first semi heuristically, but later rigorously) that somewhat surprisingly the logarithms of some generalized integers may occur as the spectrum of a fractal membrane which is a quantum fractal string. This example shows that properties of these physically motivated objects are related to the distribution of (generalized) prime numbers.

### MSC:

 11N80 Generalized primes and integers 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11M41 Other Dirichlet series and zeta functions 11N05 Distribution of primes 11S45 Algebras and orders, and their zeta functions

Zbl 1150.11003
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