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The Basel Problem: A Physicist’s Solution

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Notes

  1. The Sokhotski–Plemelj formula is a relation between generalized functions, that is, it is assumed that both sides of (20) are multiplied by a smooth function that is nonsingular in a neighborhood of the origin, and then integrated over a range of z containing the origin, and finally, the limit \(\epsilon \rightarrow 0\) is taken.

  2. Paper El30 at http://www.math.dartmouth.edu/~euler/.

References

  1. G. E. Andrews, R. Askey, and R. Roy. Special Functions. Cambridge: Cambridge University Press, 1999.

    Book  MATH  Google Scholar 

  2. B. C. Berndt. Ramanujan’s Notebooks: Part 1. New York: Springer-Verlag, 1985.

    Book  MATH  Google Scholar 

  3. K. N. Boyadzhiev . Some integrals related to the Basel problem. Scientia, series A 26 (2015), 1–13.

    MATH  Google Scholar 

  4. M. Brede. Eulers Identitäten für die Werte von \(\zeta (2n)\). Math. Semesterber. 54 (2007), 135–140.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Chamberland and A. Straub. On gamma quotients and infinite products. Adv. in Appl. Math. 51 (2013), 546–562.

    Article  MathSciNet  MATH  Google Scholar 

  6. Ó Ciaurri. Euler’s product expansion for the sine: An elementary proof. Amer. Math. Monthly 122:7 (2015), 693–695.

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Costandin. A new proof for the exact values of \(\zeta (2k)\) for \(k\in {\mathbb{N}}\). Available online at arXiv:1712.02255, 2017.

  8. D. Cvijović. Values of the derivatives of the cotangent at rational multiples of \(\pi \). Appl. Math. Lett. 22 (2009), 217–220.

    Article  MathSciNet  MATH  Google Scholar 

  9. R. Dedekind. Über ein Eulersches Integral. J. Reine Angew. Math. 45 (1852), 370–374.

    Google Scholar 

  10. W. F. Eberlein. On Euler’s infinite product for the sine. J. Math. Anal. Appl. 58:1 (1977), 147–151.

    Article  MathSciNet  MATH  Google Scholar 

  11. W. Feller. A Direct Proof of Stirling’s Formula. Amer. Math. Monthly 74:10 (1967), 1223–1225.

    Article  MathSciNet  MATH  Google Scholar 

  12. R. Feynman. The Character of Physical Law. Cambridge, MA: MIT Press, 2017.

    Book  MATH  Google Scholar 

  13. A. Ghorbanpour and M. Hatzel. Parseval’s identity and values of zeta function at even integers. Available online at arXiv:1709.09326, (2017)

  14. G. Glebov. A peculiar proof of an identity of Euler. Math. Gaz. 99:544 (2015), 139–143.

    Article  MathSciNet  Google Scholar 

  15. G. J. O. Jameson. A fresh look at Euler’s limit formula for the gamma function. Math. Gaz. 98:542 (2014), 235–242.

    Article  MathSciNet  MATH  Google Scholar 

  16. Kapil R. Shenvi Pause. Basel problem: A solution motivated by the power of a point. Amer. Math. Monthly 125:6 (2018), 558–560.

  17. D. E. Knuth and T. J. Buckholtz. Computation of tangent, Euler, and Bernoulli numbers. Math. Comp. 21 (1967), 663–688.

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Koecher. Klassische elementare Analysis. Basel: Birkhäuser, 1987.

    Book  MATH  Google Scholar 

  19. J. C. Lagarias. Euler’s constant: Euler’s work and modern developments. Bulletin Amer. Math. Soc. 50:4 (2013), 527–628.

    Article  MathSciNet  MATH  Google Scholar 

  20. N. Lord. Beyond the Basel problem: Euler’s derivation of the general formula for \(\zeta (2n)\). Math. Gaz. 98:543 (2014), 459–474.

    Article  MATH  Google Scholar 

  21. S. Moreno. A short and elementary proof of the Basel problem. College Math. J. 47:2 (2016), 134–135.

    Article  MathSciNet  MATH  Google Scholar 

  22. N. Nielsen. Handbuch der Theorie der Gammafunktion. Leipzig: B. G. Teubner, 1906.

    MATH  Google Scholar 

  23. F. Qi. Derivatives of tangent function and tangent numbers. Appl. Math. Comput. 268 (2015), 844–858.

    MathSciNet  MATH  Google Scholar 

  24. P. Ribenboim. Classical Theory of Algebraic Numbers. New York: Springer, 2001.

    Book  MATH  Google Scholar 

  25. R. Roy. Sources in the Development of Mathematics: Infinite Series and Products from the Fifteenth to the Twenty-first Century. Cambridge: Cambridge University Press, 2011.

    Book  MATH  Google Scholar 

  26. D. Salwinski. Euler’s sine product formula: An elementary proof. College Math. J. 49:2 (2018), 126–135.

    Article  MathSciNet  MATH  Google Scholar 

  27. C. E. Sandifer. Euler’s solution of the Basel problem—the longer story. In Euler at 300: An Appreciation. Washington: Mathematical Association of America, 2007.

  28. S. Siklos. A method of evaluating \(\zeta (2)\) and \(\int _0^\infty \frac{\sin {x}}{x}dx\). Math. Gaz. 102:553 (2018), 114–121.

    Article  MathSciNet  Google Scholar 

  29. R. Sitaramachandrarao and B. Davis. Some identities involving the Riemann zeta function, II. Indian J. Pure Appl. Math. 17:10 (1986), 1175–1186.

    MathSciNet  MATH  Google Scholar 

  30. G. K. Srinivasan. Dedekind’s proof of Euler’s reflection gormula via ODEs. Mathematics Newsletter 21:3 (2011), 82–83.

    MathSciNet  MATH  Google Scholar 

  31. M. Vermeeren. Modified equations and the Basel problem. Math. Intelligencer 40:2 (2018), 33–37.

    Article  MathSciNet  MATH  Google Scholar 

  32. V. S. Vladimirov. Equations of Mathematical Physics. New York: Marcel Dekker, 1971.

    MATH  Google Scholar 

  33. J. Wästlund. Summing inverse squares by Euclidean geometry. Available online at http://www.math.chalmers.se/~wastlund/Cosmic.pdf, 2010.

  34. S. Weinberg. The Quantum Theory of Fields. Volume I: Foundations. Cambridge: Cambridge University Press, 1995.

    Book  Google Scholar 

  35. G. T. Williams. A new method of evaluating \(\zeta (2n)\). Amer. Math. Monthly 60:1 (1953), 19–25.

    MathSciNet  MATH  Google Scholar 

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Acknowledgments

The work is supported by the Ministry of Education and Science of the Russian Federation. The author thanks Professor Juan Arias de Reyna for indicating several interesting references, as well as an anonymous referee for constructive comments that helped to improve the presentation.

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  1. Budker Institute of Nuclear Physics and Novosibirsk State University, 630 090, Novosibirsk, Russia

    Zurab K. Silagadze

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Silagadze, Z.K. The Basel Problem: A Physicist’s Solution. Math Intelligencer 41, 14–18 (2019). https://doi.org/10.1007/s00283-019-09902-x

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