Looking backward: From Euler to Riemann

TL;DR

This paper surveys the foundational ideas and historical development of mathematical concepts that influenced Riemann's groundbreaking work, highlighting Euler's pivotal role in shaping these fields.

Contribution

It provides a comprehensive historical overview of the mathematical subjects leading to Riemann's discoveries, emphasizing Euler's significant influence across multiple areas.

Findings

Euler's foundational role in complex analysis and topology

Historical connections between early mathematical ideas and Riemann's work

Evolution of key mathematical concepts from Euler to Riemann

Abstract

We survey the main ideas in the early history of the subjects on which Riemann worked and that led to some of his most important discoveries. The subjects discussed include the theory of functions of a complex variable, elliptic and Abelian integrals, the hypergeometric series, the zeta function, topology, differential geometry, integration, and the notion of space. We shall see that among Riemann's predecessors in all these fields, one name occupies a prominent place, this is Leonhard Euler. The final version of this paper will appear in the book \emph{From Riemann to differential geometry and relativity} (L. Ji, A. Papadopoulos and S. Yamada, ed.) Berlin: Springer, 2017.