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Some studies on arithmetical chaos in classical and quantum mechanics
 Internat. J. Modern Phys
, 1993
"... Several aspects of classical and quantum mechanics applied to a class of strongly chaotic systems are studied. The latter consists of single particles moving without external forces on surfaces of constant negative Gaussian curvature whose corresponding fundamental groups are supplied with an arithm ..."
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Several aspects of classical and quantum mechanics applied to a class of strongly chaotic systems are studied. The latter consists of single particles moving without external forces on surfaces of constant negative Gaussian curvature whose corresponding fundamental groups are supplied with an arithmetic structure. It is shown that the arithmetical features of the considered systems lead to exceptional properties of the corresponding spectra of lengths of closed geodesics (periodic orbits). The most significant one is an exponential growth of degeneracies in these geodesic length spectra. Furthermore, the arithmetical systems are distinguished by a structure that appears as a generalization of geometric symmetries. These pseudosymmetries occur in the quantization of the classical arithmetic systems as Hecke operators, which form an infinite algebra of selfadjoint operators commuting with the Hamiltonian. The statistical properties of quantum energies in the arithmetical systems have previously been identified as exceptional. They do not fit into the general scheme of random matrix theory. It is shown with the help of a simplified model for the spectral form factor
On elementary proofs of the Prime Number Theorem for arithmetic progressions, without characters.
, 1993
"... : We consider what one can prove about the distribution of prime numbers in arithmetic progressions, using only Selberg's formula. In particular, for any given positive integer q, we prove that either the Prime Number Theorem for arithmetic progressions, modulo q, does hold, or that there exist ..."
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: We consider what one can prove about the distribution of prime numbers in arithmetic progressions, using only Selberg's formula. In particular, for any given positive integer q, we prove that either the Prime Number Theorem for arithmetic progressions, modulo q, does hold, or that there exists a subgroup H of the reduced residue system, modulo q, which contains the squares, such that `(x; q; a) ¸ 2x=OE(q) for each a 62 H and `(x; q; a) = o(x=OE(q)), otherwise. From here, we deduce that if the second case holds at all, then it holds only for the multiples of some fixed integer q 0 ? 1. Actually, even if the Prime Number Theorem for arithmetic progressions, modulo q, does hold, these methods allow us to deduce the behaviour of a possible `Siegel zero' from Selberg's formula. We also propose a new method for determining explicit upper and lower bounds on `(x; q; a), which uses only elementary number theoretic computations. 1. Introduction. Define `(x) = P px log p, where p only denot...
Asymptotic Density in Combined Number Systems
"... Abstract. Necessary and sufficient conditions are found for a combination of additive number systems and a combination of multiplicative number systems to preserve the property that all partition sets have asymptotic density. These results cover and extend several special cases mentioned in the lite ..."
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Abstract. Necessary and sufficient conditions are found for a combination of additive number systems and a combination of multiplicative number systems to preserve the property that all partition sets have asymptotic density. These results cover and extend several special cases mentioned in the literature and give partial solutions to two problems in [6].
Beurling Generalized Integers with the Delone Property
, 1997
"... A set N of Beurling generalized integers consists of the unit n 0 = 1 plus the set n 1 n 2 ... of all power products of a set of generalized primes 1 ! g 1 g 2 g 3 ... with g i ! 1, with these power products arranged in increasing order and counted with multiplicity. We say that N has the Delone pro ..."
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A set N of Beurling generalized integers consists of the unit n 0 = 1 plus the set n 1 n 2 ... of all power products of a set of generalized primes 1 ! g 1 g 2 g 3 ... with g i ! 1, with these power products arranged in increasing order and counted with multiplicity. We say that N has the Delone property if there are positive constants r; R such that R n i+1 \Gamma n i r for all i 1. Any set N with the Delone property has unique factorization into irreducible elements and is therefore a subsemigroup of R + . We classify all such semigroups which are contained in the integers Z + . The set of generalized primes of any such N consists of all but finitely many primes, plus finitely many other composites.
unknown title
, 709
"... Let ω(l) = ∑ pl 1 and m, n ∈ N, n ≥ m. We calculate a formula {p ∈ P; p  () ..."
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Let ω(l) = ∑ pl 1 and m, n ∈ N, n ≥ m. We calculate a formula {p ∈ P; p  ()
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"... Vcycle optimal convergence for certain (multilevel) structured linear systems. (English summary) SIAM J. Matrix Anal. Appl. 26 (2004), no. 1, 186–214 (electronic). Summary: “In this paper we are interested in the solution by multigrid strategies of multilevel linear systems whose coefficient matric ..."
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Vcycle optimal convergence for certain (multilevel) structured linear systems. (English summary) SIAM J. Matrix Anal. Appl. 26 (2004), no. 1, 186–214 (electronic). Summary: “In this paper we are interested in the solution by multigrid strategies of multilevel linear systems whose coefficient matrices belong to the circulant, Hartley, or τ algebras or to the Toeplitz class and are generated by (the Fourier expansion of) a nonnegative multivariate polynomial f. It is well known that these matrices are banded and have eigenvalues equally distributed as f, so they are illconditioned whenever f takes the zero value; they can even be singular and need a lowrank correction. “We prove the Vcycle multigrid iteration to have a convergence rate independent of the dimension even in the presence of illconditioning. If the (multilevel) coefficient matrix has partial dimension nr at level r, r = 1,..., d, then the size of the algebraic system is N(n) = ∏d r=1 nr; O(N(n)) operations are required by our technique, and therefore the corresponding method is optimal.