An expository review of the Chebyshev-Sylvester method in prime number theory

Abstract

This paper provides a detailed expository and computational account of the elementary methods developed by P. L. Chebyshev and J. J. Sylvester to establish explicit bounds on the prime counting function. The core of the method involves replacing the M\"obius function with a finitely supported arithmetic function in the convolution identities, relating the Chebyshev function psi(x) to the summatory logarithm function T(x) = log([x]!). We present a comprehensive analysis of the various schemes proposed by Chebyshev and Sylvester, with a central focus on Sylvester's innovative iterative refinement procedure. By implementing this procedure computationally, we replicate, verify, and optimize the historical results, providing a self-contained pedagogical resource for this pivotal technique in analytic number theory.