1,601 research outputs found
Transcendental equations satisfied by the individual zeros of Riemann , Dirichlet and modular -functions
We consider the non-trivial zeros of the Riemann -function and two
classes of -functions; Dirichlet -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 . From this it follows that the
ordinate of the -th zero satisfies a transcendental equation that depends
only on . 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 -function. Our approach is a novel and simple method,
that takes into account , to numerically compute non-trivial zeros of
-functions. The method is surprisingly accurate, fast and easy to implement.
Employing these numerical solutions, in particular for the -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 -functions and the -function for the modular form based on the
Ramanujan -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
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
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 for the Riemann zeros and 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, , 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
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
- …