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.


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
Full Text: DOI arXiv


[1] Bateman, P.T., Diamond, H.G.: Asymptotic distribution of Beurling’s generalised prime numbers. In: Studies in Number Theory 6, pp. 152–212. Prentice-Hall, Englewood Cliffs, New Jersey (1969) · Zbl 0216.31403
[2] Bellisard, J.: Gap labelling theorems for Schrödinger operators. In: Waldschmidt, M. et al. (eds.) From Number Theory to Physics. Proceedings of Les Houches Meeting, France, March 1989, pp. 538–630. Springer, Berlin Heidelberg New York (1992)
[3] Berndt, B.C.: Identities involving the coefficients of a class of Dirichlet series I. Trans. Amer. Math. Soc. 137, 345–359 (1969) · Zbl 0175.32802
[4] Beurling, A.: Analyse de la loi asymptotique de la distribution des nombres premiers généralisés, I. Acta Math. 68, 255–291 (1937) · Zbl 0017.29604
[5] Bochner, S.: Some properties of modular relations. Ann. of Math. 53(2), 332–363 (1951) · Zbl 0042.32101
[6] Borel, J.-P.: Sur le prolongement des fonctions {\(\zeta\)} associées à un système de nombres premiers généralisés de Beurling. Acta Arith. 43, 273–282 (1984) · Zbl 0485.10034
[7] Connes, A.: Noncommutative Geometry Academic Press, New York (1994)
[8] Conrey, J.B., Ghosh, A.: On the Selberg class of Dirichlet series: small degrees. Duke Math. J. 72, 673–693 (1993) · Zbl 0796.11037
[9] Diamond, H.G.: The prime number theorem for Beurling generalised numbers. J. Number Theory 1, 200–207 (1969) · Zbl 0167.32001
[10] Diamond, H.G.: Asymptotic distribution of Beurling generalised integers. Illinois J. Math. 14, 12–28 (1970) · Zbl 0186.36403
[11] Diamond, H.G.: A set of generalised numbers showing Beurling’s theorem to be sharp. Illinois J. Math. 14, 29–34 (1970) · Zbl 0186.36501
[12] Diamond, H.G.: When do Beurling generalised integers have a density? J. Reine Angew. Math. 295, 22–39 (1977) · Zbl 0355.10038
[13] Diamond, H., Montgomery, H., Vorhauer, U.: Beurling primes with large oscillation. Math. Ann. 334, 1–36 (2006) · Zbl 1207.11105
[14] Edwards, H.M.: Riemann’s Zeta Function. Academic, New York (1974) (Reprinted by Dover, New York, 2001.) · Zbl 0315.10035
[15] Fuchs, L.: Partially Ordered Algebraic Systems. International Series of Monographs in Pure and Applied Mathematics, vol 28. Pergamon, New York (1963) · Zbl 0137.02001
[16] Hall, R.S.: The prime number theorem for generalised primes. J. Number Theory 4, 313–320 (1972) · Zbl 0244.10044
[17] Hall, R.S.: Theorems about Beurling’s generalised primes and the associated zeta function. Ph.D. Thesis, University of Illinois, Urbana, USA (1967)
[18] He, C.Q., Lapidus, M.L.: Generalized Minkowski content, spectrum of fractal drums, fractal strings and the Riemann zeta function. Mem. Amer. Math. Soc. 127, No. 608, 1–97 (1997) · Zbl 0877.35086
[19] Huxley, M.N.: Exponential sums and lattice points. Proc. London Math. Soc. 60, 471–475 (1990) · Zbl 0659.10057
[20] Ingham, A.E.: The Distribution of Prime Numbers, Cambridge Tracts in Math. 30, 2nd edn. Cambridge University Press, UK (1990) · Zbl 0715.11045
[21] Katz, A.: A short introduction to quasicrystallography. In: Waldschmidt, M. et al. (eds.) From Number Theory to Physics. Proceedings of Les Houches Meeting, March 1989, pp. 496–537. Springer, Berlin Heidelberg New York (1992) · Zbl 0802.52012
[22] Lagarias, J.C.: Beurling generalised integers with the Delone property. Forum Math. 11, 295–312 (1999) · Zbl 0927.11047
[23] Landau, E.: Neuer beweis des primzahlsatzes und beweis des primidealsatzes. Math. Ann. 56, 645–670 (1903) · JFM 34.0228.03
[24] Lapidus, M.L.: Fractal drums, inverse spectral problems for elliptic operators and a partial resolution of the Weyl–Berry conjecture. Trans. Amer. Math. Soc. 325, 465–529 (1991) · Zbl 0741.35048
[25] Lapidus, M.L.: Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media, and the Weyl–Berry conjecture. In: Sleeman, B.D., Jarvis, R.J. (eds.) Ordinary and Partial Differential Equations, vol. IV. Proceedings of the Twelfth International Conference, Dundee, Scotland, UK, June 1992. Pitman Research Notes in Mathematics Series, vol. 289, pp. 126–209. Longman Scientific and Technical, London (1993) · Zbl 0830.35094
[26] Lapidus, M.L.: T-duality, functional equation, and noncommutative stringy spacetime. In: Boi, L. (ed.) Geometries of Nature, Living Systems, and Human Cognition: New Interactions of Mathematics with the Natural Sciences and Humanities, pp. 3–91. World Scientific, Singapore (2005) · Zbl 1101.81090
[27] Lapidus, M.L.: In Search of the Riemann Zeros: Strings, Fractal Membranes, and Noncommutative Spacetimes. Amer. Math. Soc. (486 + (xv) pp., to appear). Providence, Rhode Island (2007)
[28] Lapidus, M.L., Maier, H.: The Riemann hypothesis and inverse spectral problems for fractal strings. J. London Math. Soc. 52(2), 15–34 (1995) · Zbl 0836.11031
[29] Lapidus, M.L., Nest, R.: Fractal membranes and the second quantization of fractal strings (in preparation)
[30] Lapidus, M.L., Nest, R.: Functional equations for zeta functions associated with quasicrystals and fractal membranes (in preparation)
[31] Lapidus, M.L., Pomerance, C.: The Riemann zeta-function and the one-dimensional Weyl–Berry conjecture for fractal drums. Proc. London Math. Soc. 66(3), 41–69 (1993) · Zbl 0788.34083
[32] Lapidus, M.L., Pomerance, C.: Counterexamples to the modified Weyl–Berry conjecture on fractal drums. Proc. Cambridge Philos. Soc. 199, 167–178 (1996) · Zbl 0858.58052
[33] Lapidus, M.L., van Frankenhuysen, M.: Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions. Birkhäuser, Boston, Massachusetts (2000) · Zbl 0981.28005
[34] Lapidus, M.L., van Frankenhuijsen, M.: Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings. Springer Monographs in Mathematics (464+(xxiv)pp). Springer, Berlin Heidelberg New York (2006) · Zbl 1119.28005
[35] Lapidus, M.L., van Frankenhuijsen, M.: Fractality, self-similarity and complex dimensions. In: Lapidus, M. L., van Frankenhuysen, M. (eds.) Fractal Geometry and Number Theory: A Jubilee of Benoit Mandelbrot. Part 1. Proceedings of Symposia in Pure Mathematics, vol. 72, pp. 349–372. Amer. Math. Soc. Providence, Rhode Island (2004) · Zbl 1196.11100
[36] Malliavin, P.: Sur le reste de la loi asymptotique de répartition des nombres premiers généralisés de Beurling. Acta Math. 106, 281–298 (1961) · Zbl 0102.28204
[37] Nyman, B.: A general prime number theorem. Acta Math. 81, 299–307 (1949) · Zbl 0034.02402
[38] Ram Murty, M.: Selberg’s conjectures and Artin L-functions. Bull. Amer. Math. Soc. 31, 1–14 (1994) · Zbl 0805.11062
[39] Ryavec, C.: The analytic continuation of Euler products with applications to asymptotic formulae. Illinois J. Math. 17, 608–616 (1973) · Zbl 0266.10042
[40] Selberg, A.: Old and new conjectures about a class of Dirichlet series. In: Selberg, A. (ed.) Collected papers, vol II, pp. 47–63. Springer, Berlin Heidelberg New York (1991)
[41] Senechal, M.: Quasicrystals and Geometry Cambridge University Press, UK (1995)
[42] Titchmarsh, E.C.: The Theory of Functions, 2nd edn. Oxford University Press, London, UK (1986) · Zbl 0601.10026
[43] Titchmarsh, E.C.: The Theory of the Riemann Zeta-function, 2nd edn. Oxford University Press, London, UK (1986) · Zbl 0601.10026
[44] Zhang, W.-B.: Density and O-density of Beurling generalised integers. J. Number Theory 30, 120–139 (1988) · Zbl 0677.10032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.