El Logaritmo Integral: N\'umeros primos y algo m\'as

TL;DR

This paper explores the historical development and significance of the logarithmic integral function in 19th-century mathematics, highlighting its connections to prime number distribution and contributions from key mathematicians like Gauss and Bessel.

Contribution

It provides a detailed historical analysis of the logarithmic integral's role in mathematics, emphasizing previously underappreciated contributions of Bessel and clarifying its influence on prime number theory.

Findings

Established the timeline of logarithmic integral developments.

Highlighted Bessel's collaboration with Gauss.

Clarified the integral's role in prime number distribution.

Abstract

We show the relevance of the logarithmic integral function in the development of mathematics in the first half of the 19th century. Its importance involved first level mathematicians such as Euler, Gauss, Bessel, Riemann. Our perspective is the result of a detailed study of the original sources. We manage to establish the timeline of how the advances took place. In particular, our study vindicates the contributions of Bessel (in collaboration with Gauss). The logarithmic integral in Gauss's mind played a fundamental role in several fields such as complex analysis (Cauchy's theorem), numerical methods (Gaussian quadrature) as well as its best known relation with the distribution of prime numbers. We study this last aspect in detail starting from the works by Legendre and Tchebycheff up to Riemann.