Zeta Function Zeros, Powers of Primes, and Quantum Chaos

Abstract

We present a numerical study of Riemann's formula for the oscillating part of the density of the primes and their powers. The formula is comprised 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 powers of primes and demonstrate explicitly that the relative line intensities are correct. We then derive a Gaussian sum rule for Riemann's formula. This is used to analyze the numerical convergence of the truncated series. The connections to quantum chaos and semiclassical physics are discussed.

References in corpus (3)

• On the Role of Non-Periodic Orbits in The Semiclassical Quantization of the Truncated Hyperbola Billiard
• Closed almost-periodic orbits in semiclassical quantization of generic polygons
• Nonperiodic Orbit Sums in Weyl's Expansion for Billiards

Cited by in corpus (10)

• Physics of the Riemann Hypothesis
• Hidden structure in the randomness of the prime number sequence?
• Stability and Dynamics of Complex Order Fractional Difference Equations
• Riemann zeta zeros and prime number spectra in quantum field theory
• Hilbert-Pólya Conjecture, Zeta-Functions and Bosonic Quantum Field Theories
• Pseudo Random test of prime numbers
• Fractal in the statistics of Goldbach partition
• Jacob's Ladder: Prime numbers in 2d
• Functional equations for regularized zeta-functions and diffusion processes
• Unconditional and Conditional Large Gaps between the zeros of the Riemann Zeta-Function