From The Editors
4 September 2025
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Volume 5, Issue 1 (2025)Read More
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Articles & Notes
4 September 2025
How Euler Could Have Done It: Euler and a direct proof for the functional equation for the Riemann zeta-function
We present a chain of argumentation for a direct proof of the functional equation for the Riemann ζ–function that Euler could have presented.Current Articles
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Translation & Commentary
19 August 2021
Euler's three-body problem
In physics and astronomy, Euler's three-body problem is to solve for the motion of a body that is acted upon by the gravitational field of two other bodies. This problem is named after Leonhard Euler (1707-1783), who discussed it in memoirs published in the 1760s. In these publications, Euler found that the parameter that controls the relative distances among three collinear bodies is given by a quintic equation. Later on, in 1772, Lagrange dealt with the same problem, and demonstrated that for any three masses with circular orbits, there are two special constant-pattern solutions, one where the three bodies remain collinear, and the other where the bodies occupy the vertices of two equilateral triangles. Because of their importance, these five points became known as Lagrange points. The quintic equation found by Euler for the relative distances among the collinear bodies was also found later by Lagrange, and because of that, Euler has also been given credit for the discovery of the three collinear Lagrange points. A practical application of the collinear points for satellite location is also presented.From The Editors
4 September 2025
Comprehensive Conversations
An introduction to the contents in Volume 5 (Issue 1) of Euleriana.Articles & Notes
4 September 2025
How Euler Could Have Done It: Euler and a direct proof for the functional equation for the Riemann zeta-function
We present a chain of argumentation for a direct proof of the functional equation for the Riemann ζ–function that Euler could have presented.Articles & Notes
19 September 2022
Basel Problem: Historical perspective and further proofs from stochastic processes
In this note, we offer a historical perspective on solutions of the Basel problem. In particular, we have a closer look at some of the less famous results by Euler E41 and provide a review of a selection of the assemblage of earlier proofs. Moreover, we show how to generate further proofs using Karhunen-Lo\`{e}ve expansions of stochastic processes.Articles & Notes
19 August 2021
Euler's Miracle
This article features some genuine Eulerian magic. In 1748, Leonhard Euler considered a modification of the harmonic series in which negative signs were attached to various terms by a rule that was far from self-evident. With his accustomed flair, he determined its sum, and the result was utterly improbable. There are a few occasions in mathematics when the term “breathtaking” is not too strong. This is one of them.Translation & Commentary
4 September 2025