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Riemann equation for prime number diffusion.

Wen Chen1, Yingjie Liang1

  • 1Institute of Soft Matter Mechanics, College of Mechanics and Materials, Hohai University, No. 1 XiKang Road, Nanjing, Jiangsu 210098, China.

Chaos (Woodbury, N.Y.)
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This summary is machine-generated.

This study introduces the Riemann diffusion equation to model prime number distribution using physics of diffusion. The equation

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Area of Science:

  • Number Theory
  • Mathematical Physics
  • Statistical Mechanics

Background:

  • The distribution of prime numbers is a fundamental problem in number theory.
  • Classical methods for prime number distribution lack a physical interpretation.
  • The Riemann hypothesis remains unproven, motivating new analytical approaches.

Purpose of the Study:

  • To propose a novel partial differential equation, the Riemann diffusion equation, for prime number distribution.
  • To interpret prime number distribution through the lens of diffusion physics.
  • To analyze the properties of the Riemann diffusion equation and its solution.

Main Methods:

  • Formulating the Riemann diffusion equation as a partial differential equation.
  • Deriving the analytical solution, which corresponds to the Riemann representation.
  • Investigating the position-dependent diffusivity and its behavior.
  • Obtaining an approximate solution and comparing it with the established Riemann representation.

Main Results:

  • The diffusion coefficient exhibits position-dependent diffusivity, closely following a power law.
  • An approximate solution of the Riemann diffusion equation accurately predicts prime number distribution.
  • The scale-free property of prime number distribution is described by a power law with an exponent of 1.0169.
  • The fractal characteristic of prime number distribution is revealed.

Conclusions:

  • The Riemann diffusion equation provides a novel physical interpretation for prime number distribution.
  • The study demonstrates a connection between diffusion processes and number theory.
  • The findings offer new insights into the scale-free and fractal nature of prime numbers.