A Quantum Instrument Plays the Music of Primes ⚡ экспресс
In numerical experiments, a symmetric potential was constructed for the one-dimensional stationary Schrödinger equation, such that its eigenvalue spectrum coincides with the imaginary parts of the first non-trivial zeros of the Riemann zeta function. The potential was generated by successively adding correction functions to an initial smooth approximation based on the Riemann–von Mangoldt formula for the asymptotic number of zeros. It was found that the correction functions follow a clear pattern almost entirely determined by the error in this formula. This pattern reveals the nature of the fractal pattern of the potential previously observed by Wu and Sprung, and indicates that the corrections reflect the fine structure of the prime number distribution.
Physicists have created a quantum analogue of a musical instrument. Its “notes”—the allowed energy levels—precisely replicate the first numbers from a set that governs the distribution of primes. These numbers are the imaginary parts of the zeros of the famous Riemann zeta function, a mathematical object inextricably linked to prime numbers.
The instrument didn’t sound right at first. Scientists began with a smooth approximation, then added patch-like corrections. To their surprise, all corrections obeyed a simple rule—they were determined by the error in the well-known prime-counting formula (the Riemann–von Mangoldt formula). It was this simple mechanism that generated the fractal pattern previously noticed in the potential’s structure. Thus, the quantum spectrum reflected the hidden harmony of number theory—much like the hydrogen atom produces a strict set of spectral lines. The link between quantum disorder and arithmetic is no longer just a metaphor; it’s becoming a tool for assaulting a great mathematical problem.
🎯 Prime numbers are sometimes called the “atoms of arithmetic” because they serve as the building blocks from which all whole numbers are constructed.