Zeta function zeros, powers of primes, and quantum chaos.

We present a numerical study of Riemann's formula for the oscillating part of the density of the primes and their integer powers. The formula consists of an infinite series of oscillatory terms, one for each zero of the zeta function on the critical line, and was derived by Riemann in his paper on primes, assuming the Riemann hypothesis. We show that high-resolution spectral lines can be generated by the truncated series at all integer powers of primes and demonstrate explicitly that the relative line intensities are correct.

Number fluctuation and the fundamental theorem of arithmetic.

We consider N bosons occupying a discrete set of single-particle quantum states in an isolated trap. Usually, for a given excitation energy, there are many combinations of exciting different number of particles from the ground state, resulting in a fluctuation of the ground state population. As a counterexample, we take the quantum spectrum to be logarithms of the prime number sequence, and using the fundamental theorem of arithmetic, find that the ground state fluctuation vanishes exactly for all excitations.