By M. Aizenman (Chief Editor)

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A heuristic preamble. We begin to give a heuristic, but elementary, motivation for our dimension formula, postponing for the moment the specification of the underlying structure. Let the selfadjoint operator H0 be a reference Hamiltonian for a Quantum Statistical Mechanics system, generating the evolution on an operator algebra A. H0 may be thought to correspond to the Laplacian on a compact Riemann manifold. We write any other Hamiltonian H as a perturbation H = H0 + P of H0 . We assume a supersymmetric structure, essentially the existence of a Dirac operator D (an odd square root of H ) that implements an odd derivation of A.

Elementary Number Theory. , 1958 13. : Topics in Multiplicative Number Theory. Lecture Notes in Mathematics 227, Berlin– Heidelberg: Springer-Verlag, 1971 14. : An Introduction to Algebraic Number Theory. New York, London: Plenum Press, 1990 15. : Momente der Klassenzahlen binärer quadratischer Formen mit ganzalgebraischen Koeffizienten. Acta Arith. 1, 43–77 (1995) 16. : The limit distribution of a number theoretic function arising from a problem in statistical mechanics. To appear in J. Number Th.

32 J. Kallies, A. Özlük, M. Peter, C. 5. ve (N ) ∼ 3 log 2 2 N , π2 as N −→ ∞. Proof. We first obtain a relation between ve (N ) and r(N ). 1, 1 ε+ (ω)k < tr ε+ (ω)k < ε+ (ω)k + . 3, r (N − 1/2)1/k < ve (N ) < k<2 log N r N 1/k . 4, r N 1/k ∼ 3 log 2 2/k N π2 N, as N −→ ∞, if 2/k ≤ 1. We thus have r (N − 1/2)1/k < 2≤k<2 log N r N 1/k N log N = o(N 2 ). 4, ve (N ) ∼ 3 log 2 2 N , π2 as desired. Notice that if (N ) is to have the desired asymptotic property, then all the main contribution comes from 2vo (N ).

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