The Modern History of the Basel Problem

The Basel problem refers to the problem of determining a closed form for the infinite series 1/1² + 1/2² + ⋯. If we consider what mathematical results have the most peer-reviewed papers devoted to new ways of proving such results, Euler's formula 1/1² + 1/2² + ⋯ = π²/6 is certainly among the top of such results. This motivates our historical study of peer-reviewed papers based on proofs of Euler's formula, and we introduce what appears to be the most comprehensive and up-to-date and exhaustive catalogue of peer-reviewed, published papers in the 20th and 21st centuries devoted to or mainly concerning proofs of 1/1² + 1/2² + ⋯ = π²/6. We show how the modern history of the Basel problem relates to many of Euler's original publications, including E41, E61, E63, E72, E421, and E592, and we categorize modern solutions to the Basel problem according to the techniques involved, and we apply our findings as a basis for our historical arguments and analyses.