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

Abstract

This paper gives a self-contained expository and computational account of the elementary methods developed by P.L. Chebyshev and J.J. Sylvester to obtain explicit bounds for the distribution of prime numbers. The method replaces the Mobius function in a basic convolution identity by finitely supported arithmetic functions, or schemes, and relates the resulting auxiliary functions to the Chebyshev function psi(x) through the summatory logarithm T(x)=log([x]!). We examine a number of schemes introduced by Chebyshev and Sylvester, with particular emphasis on Sylvester's iterative refinement procedure. For a fixed choice of terms, this procedure leads to an explicit two-dimensional affine recurrence for upper and lower bounds on psi(x), whose fixed point can be analyzed by elementary linear algebra. We then implement the procedure computationally and introduce a simple one-parameter selection rule, the rho-rule, for choosing which terms to retain in the iteration. Numerical exploration of the historical schemes reproduces Sylvester's bounds and, in several cases, yields modest improvements of the resulting constants. The accompanying Python implementation is publicly available.

Revised version. Corrected the upper-bound hypothesis by adding the required global nonnegativity of E(x). Added discussion of related work by Camargo-Martin and Diamond-Erdos, cited an earlier 2018 version of these notes, and updated the acknowledgements, bibliography, and abstract. Minor expository revisions throughout. 21 pages, 20 figures