  We exhibit very small eigenvalues of the quadratic form associated to the
Weil explicit formulas restricted to test functions whose support is within a
fixed interval with upper bound S. We show both numerically and conceptually
that the associated eigenvectors are obtained by a simple arithmetic operation
of finite sum using prolate spheroidal wave functions associated to the scale
S. Then we use these functions to condition the canonical spectral triple of
the circle of length L=2 Log(S) in such a way that they belong to the kernel of
the perturbed Dirac operator. We give numerical evidence that, when one varies
L, the low lying spectrum of the perturbed spectral triple resembles the low
lying zeros of the Riemann zeta function. We justify conceptually this result
and show that, for each eigenvalue, the coincidence is perfect for the special
values of the length L of the circle for which the two natural ways of
realizing the perturbation give the same eigenvalue. This fact is tested
numerically by reproducing the first thirty one zeros of the Riemann zeta
function from our spectral side, and estimate the probability of having
obtained this agreement at random, as a very small number whose first fifty
decimal places are all zero. The theoretical concept which emerges is that of
zeta cycle and our main result establishes its relation with the critical zeros
of the Riemann zeta function and with the spectral realization of these zeros
obtained by the first author.
