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Everything about Basel problem

Sum of inverse squares of natural numbers

The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate fame when he was twenty-eight. Euler generalised the problem considerably, and his ideas were taken up more than a century later by Bernhard Riemann in his seminal 1859 paper "On the Number of Primes Less Than a Given Magnitude", in which he defined the Riemann zeta function and proved its basic properties. The problem is named after the city of Basel, hometown of Euler as well as of the Bernoulli family who unsuccessfully attacked the problem. Wikipedia →

🧭 Start here: a 6-step reading path

  1. Read the overview: Basel problemA 5-minute orientation
  2. Skim a recent paper: Clustering as a Solution to Environmental Problem Shifting? The Case of the Basel–Rotterdam–Stockholm Synergies ProcessInfluential work since 2022
  3. Pick up a classic book: Hoffnung für unsere ZukunftThe most-reprinted book on the subject
  4. Read the landmark paper: The Basel Problem as a Telescoping SeriesCited 13 times
  5. Go sideways: Clausen functionA closely related topic
  6. Go sideways: Dirichlet beta functionA closely related topic

📖 Definitions

No dictionary entry for this exact phrase.

🗂️ The subject at a glance

  1. Euler's approach
  2. The Riemann zeta function
  3. A proof using Euler's formula and L'Hôpital's rule
  4. A proof using Fourier series
  5. Another proof using Parseval's identity
  6. Proof using differentiation under the integral sign
  7. Cauchy's proof
  8. Proof assuming Weil's conjecture on Tamagawa numbers
  9. Geometric proof
  10. Other identities

🏆 Landmark papers (most cited)

🆕 Influential since 2022

📚 Books worth reading

Ranked by number of editions on Open Library, a rough measure of how lasting a book is.

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