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 random-matrix 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 Riemann-Siegel 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 Riemann’s hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form sn = 1/2+iλn. An improvement of our previous construction to prove the RH is presented by implementing the Hilbert-Polya proposal and furnishing the Fractal Supersymmetric Quantum Mechanical (SUSY-QM) model whose spectrum reproduces the imaginary parts of the zeta zeros. We model the fractal fluctuations of the smooth Wu-Sprung potential ( that capture the average level density of zeros � ) by recurring to a weighted superposition of Weierstrass functions p W (x, p, D) and where the summation has to be performed over all primes p in order to recapture the connection between the distribution of zeta zeros and prime numbers. We proceed next with the construction of a smooth version of the fractal QM wave equation by writing an ordinary Schroedinger equation whose fluctuating potential (relative to the smooth Wu-Sprung potential) has the same functional form as the fluctuating part of the level density of zeros. The second approach to prove the RH relies on the existence of a continuous family of scaling-like operators involving the Gauss-Jacobi theta series. An explicit trace formula related to a superposition of eigenfunctions of these scaling-like operators is defined. If the trace relation is satisfied this could be another test of the Riemann Hypothesis 1

The Riemann’s hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form s =1/2+iλn. Hilbert-Polya argued that if a Hermitian operator exists whose eigenvalues are the imaginary parts of the zeta zeros, λn’s, then the RH is true. In this paper a fractal supersymmetric quantum mechanical (SUSY-QM) model is proposed to prove the RH. It is based on a quantum inverse scattering method related to a fractal potential given by a Weierstrass function (continuous but nowhere differentiable) that is present in the fractal analog of the CBC (Comtet, Bandrauk, Campbell) formula in SUSY QM. It requires using suitable fractal derivatives and integrals of irrational order whose parameter β is one-half the fractal dimension of the Weierstrass function. An ordinary SUSY-QM oscillator is constructed whose eigenvalues are of the form λn = nπ, and which coincide with the imaginary parts of the zeros of the function sin(iz). This sine function obeys a trivial analog of the RH. A review of our earlier proof of the RH based on a SUSY QM model whose potential is related to the Gauss-Jacobi theta series is also included. The spectrum is given by s(1 − s) which is real in the critical line (location of the nontrivial zeros) and in the real axis (location of the trivial zeros). 1

Two methods to prove the Riemann Hypothesis are presented. One is based on the modular properties of Θ (theta) functions and the other on the Hilbert–Polya proposal to find an operator whose spectrum reproduces the ordinates ρn (imaginary parts) of the zeta zeros in the critical line: sn = 1 +iρn. A detailed analysis of a one-dimensional Dirac-like 2 operator with a potential V (x) is given that reproduces the spectrum of energy levels En = ρn, when the boundary conditions ΨE(x = −∞) =±ΨE(x =+∞) areimposed. Such potential V (x) is derived implicitly from the relation x = x(V) = π (dN (V)/dV), 2 where the functional form of N (V) is given by the full-fledged Riemann–von Mangoldt counting function of the zeta zeros, including the fluctuating as well as the O(E−n) terms. The construction is also extended to self-adjoint Schroedinger operators. Crucial is the introduction of an energy-dependent cut-off function Λ(E). Finally, the natural quantization of the phase space areas (associated to nonperiodic crystal-like structures) in integer multiples of π follows from the Bohr–Sommerfeld quantization conditions of Quantum Mechanics. It allows to find a physical reasoning why the average density of the primes distribution for very large x(O ( 1)) has a one-to-one correspondence with log x the asymptotic limit of the inverse average density of the zeta zeros in the critical line suggesting intriguing connections to the renormalization group program.

The Riemann’s hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form sn = 1/2 + iλn. By constructing a continuous family of scaling-like operators involving the Gauss-Jacobi theta series and by invoking a novel CT-invariant Quantum Mechanics, involving a judicious charge conjugation C and time reversal T operation, we show why the Riemann Hypothesis is true. An infinite family of theta series and their Mellin transform leads to the same conclusions. 1