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17
The Riemann Zeros and Eigenvalue Asymptotics
 SIAM Rev
, 1999
"... Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many feat ..."
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Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many features of H cl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the t n have a similar structure to those of the semiclassical En ; in particular, they display randommatrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the t n can be computed accurately from formulae with quantum analogues. The RiemannSiegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian H cl = XP. Key words. spectral asymptotics, number theory AMS subject classifications. 11M26, 11M06, 35P20, 35Q40, 41A60, 81Q10, 81Q50 PII. S0036144598347497 1.
The Explicit Formula and the conductor operator
, 1999
"... I give a new derivation of the Explicit Formula for an arbitrary number field and abelian DirichletHecke character, which treats all primes in exactly the same way, whether they are discrete or archimedean, and also ramified or not. This is followed with a local study of a Hilbert space operator, t ..."
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Cited by 4 (3 self)
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I give a new derivation of the Explicit Formula for an arbitrary number field and abelian DirichletHecke character, which treats all primes in exactly the same way, whether they are discrete or archimedean, and also ramified or not. This is followed with a local study of a Hilbert space operator, the “conductor operator”, which is expressed as H = log(x)+log(y) (where x and y are Fourier dual variables on a νadic completion of the number field). I also study the commutator operator K = i [log(y), log(x)] (which shares with H the property of complete dilation invariance, and turns out to be bounded), as well as the higher commutator operators. The generalized eigenvalues of these operators are given by the derivatives on the critical line of the TateGel’fandGraev Gamma function, which itself is in fact closely related to the additive Fourier Transform viewed in multiplicative terms. This spectral analysis is thus a natural continuation to Tate’s Thesis in its local aspects.
Eigenvalue Density, Li’s Positivity, and the Critical Strip
, 2009
"... We rewrite the zerocounting formula within the critical strip of the Riemann zeta function as a cumulative density distribution; this subsequently allows us to derive an integral expression for the Li coefficients associated with the Riemann ξfunction and, in particular, indicate that their positi ..."
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We rewrite the zerocounting formula within the critical strip of the Riemann zeta function as a cumulative density distribution; this subsequently allows us to derive an integral expression for the Li coefficients associated with the Riemann ξfunction and, in particular, indicate that their positivity criterion is obeyed, whereby entailing the criticality of the nontrivial zeros. We also
Scale invariant correlations and the distribution of prime numbers
, 903
"... Negative correlations in the distribution of prime numbers are found to display a scale invariance. This occurs in conjunction with a nonstationary behavior. We compare the prime number series to a type of fractional Brownian motion which incorporates both the scale invariance and the nonstationary ..."
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Negative correlations in the distribution of prime numbers are found to display a scale invariance. This occurs in conjunction with a nonstationary behavior. We compare the prime number series to a type of fractional Brownian motion which incorporates both the scale invariance and the nonstationary behavior. Interesting discrepancies remain. The scale invariance also appears to imply the Riemann hypothesis and we study the use of the former as a test of the latter. 1 Scale Invariance The distribution of prime numbers has been a source of fascination for mathematicians for centuries. Of interest to physicists is the fact that the distribution of prime numbers exhibits certain characteristics that are usually associated with chaotic and complex systems. The data on the prime number series is plentiful and a surprisingly rich structure is uncovered when this data is probed in various ways [1]. The closely associated zeros of the Riemann zeta function have been studied even more intensely from a physics viewpoint [2]. We shall show that prime number differences display a scale invariance in their correlations. The differences also display a nonstationary behavior that coexists with the scale invariance. We shall identify a random process, fractional Brownian motion of a lesser known type, that displays these and other properties of the prime number series. But there are still quantitative differences which suggest that a further generalization of fractional Brownian motion remains to be developed. The resolution of this puzzle could have practical value for descriptions of complex systems in physics and economics. In addition we shall describe how a breakdown of the Riemann hypothesis would not be compatible with the preservation of scale invariance.
QUANTUM COMPUTING AND ZEROES OF ZETA FUNCTIONS WIM VAN DAM
, 2004
"... Abstract. A possible connection between quantum computing and Zeta functions of finite field equations is described. Inspired by the spectral approach to the Riemann conjecture, the assumption is that the zeroes of such Zeta functions correspond to the eigenvalues of finite dimensional unitary opera ..."
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Abstract. A possible connection between quantum computing and Zeta functions of finite field equations is described. Inspired by the spectral approach to the Riemann conjecture, the assumption is that the zeroes of such Zeta functions correspond to the eigenvalues of finite dimensional unitary operators of natural quantum mechanical systems. The notion of universal, efficient quantum computation is used to model the desired quantum systems. Using eigenvalue estimation, such quantum circuits would be able to approximately count the number of solutions of finite field equations with an accuracy that does not appear to be feasible with a classical computer. For certain equations (Fermat hypersurfaces) it is show that one can indeed model their Zeta functions with efficient quantum algorithms, which gives some evidence in favor of the proposal of this article.
mean spectral density as the Riemann zeros
, 2011
"... For the classical hamiltonian (x +1/x)(p +1/p), with position x and conjugate momentum p,allorbitsarebounded. Afterasymmetrization,thecorresponding quantum integral equation possesses a family of selfadjoint extensions: compact operators on the entire positive x axis, labelled by an angle α specify ..."
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For the classical hamiltonian (x +1/x)(p +1/p), with position x and conjugate momentum p,allorbitsarebounded. Afterasymmetrization,thecorresponding quantum integral equation possesses a family of selfadjoint extensions: compact operators on the entire positive x axis, labelled by an angle α specifying the boundary condition at the origin, with a discrete spectrum of real energies E.Onthecylinder{− ∞ < E < ∞,0 � α<2π},thereisasingleeigencurvein the form of a helix winding clockwise. The rise between successive windings gets sharper as the scaled Planck’s constant decreases. This behaviour can be understood semiclassically. The first two terms of the asymptotic eigenvalue density are the same as those for the density of heights of the Riemann zeros. PACS numbers: 02.30.Mv, 02.30.Tb, 03.65.Ge, 03.65.Sq (Some figures in this article are in colour only in the electronic version)
An essay on the Riemann Hypothesis
"... Abstract The Riemann hypothesis is, and will hopefully remain for a long time, a great motivation to uncover and explore new parts of the mathematical world. After reviewing its impact on the development of algebraic geometry we discuss three strategies, working concretely at the level of the explic ..."
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Abstract The Riemann hypothesis is, and will hopefully remain for a long time, a great motivation to uncover and explore new parts of the mathematical world. After reviewing its impact on the development of algebraic geometry we discuss three strategies, working concretely at the level of the explicit formulas. The first strategy is “analytic ” and is based on Riemannian spaces and Selberg’s work on the trace formula and its comparison with the explicit formulas. The second is based on algebraic geometry and the RiemannRoch theorem. We establish a framework in which one can transpose many of the ingredients of the Weil proof as reformulated by Mattuck, Tate and Grothendieck. This framework is elaborate and involves noncommutative geometry, Grothendieck toposes and tropical geometry. We point out the remaining difficulties and show that RH gives a strong motivation to develop algebraic geometry in the emerging world of characteristic one. Finally we briefly discuss a third strategy based on the development of a suitable “Weil cohomology”, the role of Segal’s Γrings and of topological cyclic homology as a model for “absolute algebra ” and as a cohomological tool. 1
QUANTUM COMPUTING AND ZEROES OF ZETA FUNCTIONS WIM VAN DAM
, 2004
"... Abstract. A possible connection between quantum computing and Zeta functions of finite field equations is described. Inspired by the spectral approach to the Riemann conjecture, the assumption is that the zeroes of such Zeta functions correspond to the eigenvalues of finite dimensional unitary opera ..."
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Abstract. A possible connection between quantum computing and Zeta functions of finite field equations is described. Inspired by the spectral approach to the Riemann conjecture, the assumption is that the zeroes of such Zeta functions correspond to the eigenvalues of finite dimensional unitary operators of natural quantum mechanical systems. The notion of universal, efficient quantum computation is used to model the desired quantum systems. Using eigenvalue estimation, such quantum circuits would be able to approximately count the number of solutions of finite field equations with an accuracy that does not appear to be feasible with a classical computer. For certain equations (Fermat hypersurfaces) it is show that one can indeed model their Zeta functions with efficient quantum algorithms, which gives some evidence in favor of the proposal of this article.