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Zeta function zeros, powers of primes, and quantum chaos
Jamal Sakhr1, Rajat K Bhaduri, Brandon P van Zyl
1Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada L8S 4M1.
Summary
This study numerically validates Riemann's formula for prime number distribution using spectral line analysis. The findings confirm the formula's accuracy for prime powers and explore its connections to quantum chaos.
Area of Science:
- Number Theory
- Mathematical Physics
Background:
- Riemann's formula describes the oscillating part of prime number density.
- The formula relies on the zeros of the zeta function, assuming the Riemann hypothesis.
Purpose of the Study:
- To numerically investigate Riemann's formula for prime numbers and their powers.
- To analyze the convergence properties of the truncated series and its relation to quantum chaos.
Main Methods:
- Numerical computation of Riemann's formula using truncated series.
- Generation and analysis of spectral lines for prime powers.
- Derivation and application of a Gaussian sum rule.
Main Results:
- High-resolution spectral lines were generated for prime powers.
- The relative line intensities were shown to be accurate.
- Numerical convergence of the truncated series was analyzed using the sum rule.
Conclusions:
- The study provides numerical evidence supporting Riemann's formula.
- The findings highlight connections between number theory and quantum chaos.