Several Elementary Estimates of Prime Number Distribution and Their Improvements

Abstract

This paper systematically reviews elementary estimation methods and their improved techniques for prime number distribution, focusing on the theoretical foundations and applied value of the Euler sieve, Chebyshev's theorem, and elementary proofs of the prime number theorem. It also analyzes the latest progress in improvements based on analytical methods and numerical optimization. By reviewing the mathematical principles and historical development of elementary methods, this paper illustrates their important role in revealing the laws governing prime number distribution, while also pointing out their limitations in accuracy and complexity. The article also speaks of possibilities of using contemporary elementary methods for building algorithms and cryptography and exploring the future of combining analysis tools and computational machinery. This article provides theoretical resources and practicable observations for future studies and interdisciplinary applications pertaining to prime number distribution.