1,601 research outputs found

    Transcendental equations satisfied by the individual zeros of Riemann ζ\zeta, Dirichlet and modular LL-functions

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    We consider the non-trivial zeros of the Riemann ζ\zeta-function and two classes of LL-functions; Dirichlet LL-functions and those based on level one modular forms. We show that there are an infinite number of zeros on the critical line in one-to-one correspondence with the zeros of the cosine function, and thus enumerated by an integer nn. From this it follows that the ordinate of the nn-th zero satisfies a transcendental equation that depends only on nn. Under weak assumptions, we show that the number of solutions of this equation already saturates the counting formula on the entire critical strip. We compute numerical solutions of these transcendental equations and also its asymptotic limit of large ordinate. The starting point is an explicit formula, yielding an approximate solution for the ordinates of the zeros in terms of the Lambert WW-function. Our approach is a novel and simple method, that takes into account argL\arg L, to numerically compute non-trivial zeros of LL-functions. The method is surprisingly accurate, fast and easy to implement. Employing these numerical solutions, in particular for the ζ\zeta-function, we verify that the leading order asymptotic expansion is accurate enough to numerically support Montgomery's and Odlyzko's pair correlation conjectures, and also to reconstruct the prime number counting function. Furthermore, the numerical solutions of the exact transcendental equation can determine the ordinates of the zeros to any desired accuracy. We also study in detail Dirichlet LL-functions and the LL-function for the modular form based on the Ramanujan τ\tau-function, which is closely related to the bosonic string partition function.Comment: Matches the version to appear in Communications in Number Theory and Physics, based on arXiv:1407.4358 [math.NT], arXiv:1309.7019 [math.NT], and arXiv:1307.8395 [math.NT

    Periodic orbit spectrum in terms of Ruelle--Pollicott resonances

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    Fully chaotic Hamiltonian systems possess an infinite number of classical solutions which are periodic, e.g. a trajectory ``p'' returns to its initial conditions after some fixed time tau_p. Our aim is to investigate the spectrum tau_1, tau_2, ... of periods of the periodic orbits. An explicit formula for the density rho(tau) = sum_p delta (tau - tau_p) is derived in terms of the eigenvalues of the classical evolution operator. The density is naturally decomposed into a smooth part plus an interferent sum over oscillatory terms. The frequencies of the oscillatory terms are given by the imaginary part of the complex eigenvalues (Ruelle--Pollicott resonances). For large periods, corrections to the well--known exponential growth of the smooth part of the density are obtained. An alternative formula for rho(tau) in terms of the zeros and poles of the Ruelle zeta function is also discussed. The results are illustrated with the geodesic motion in billiards of constant negative curvature. Connections with the statistical properties of the corresponding quantum eigenvalues, random matrix theory and discrete maps are also considered. In particular, a random matrix conjecture is proposed for the eigenvalues of the classical evolution operator of chaotic billiards

    Riemann zeros, prime numbers and fractal potentials

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    Using two distinct inversion techniques, the local one-dimensional potentials for the Riemann zeros and prime number sequence are reconstructed. We establish that both inversion techniques, when applied to the same set of levels, lead to the same fractal potential. This provides numerical evidence that the potential obtained by inversion of a set of energy levels is unique in one-dimension. We also investigate the fractal properties of the reconstructed potentials and estimate the fractal dimensions to be D=1.5D=1.5 for the Riemann zeros and D=1.8D = 1.8 for the prime numbers. This result is somewhat surprising since the nearest-neighbour spacings of the Riemann zeros are known to be chaotically distributed whereas the primes obey almost poisson-like statistics. Our findings show that the fractal dimension is dependent on both the level-statistics and spectral rigidity, Δ3\Delta_3, of the energy levels.Comment: Five postscript figures included in the text. To appear in Phys. Rev.

    A central limit theorem for the zeroes of the zeta function

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    On the assumption of the Riemann hypothesis, we generalize a central limit theorem of Fujii regarding the number of zeroes of Riemann's zeta function that lie in a mesoscopic interval. The result mirrors results of Soshnikov and others in random matrix theory. In an appendix we put forward some general theorems regarding our knowledge of the zeta zeroes in the mesoscopic regime.Comment: 22 pages. Incorporates referees suggestions. Contains minor corrections to published versio
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