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In The Shadow of Euler’s Greatness: Adventures in the Rediscovery of an Intriguing Sum

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Notes

  1. Euler himself is thought to have given at least four different proofs in his lifetime [46].

  2. A recent discussion of this seminal paper has been given by Ryotaro Harada [32].

  3. Only much later, in 2015, Kunle Adegoke showed that every linear Euler sum can be decomposed into a linear combination of Tornheim’s harmonic double series [1].

References

  1. K. Adegoke. On generalized harmonic numbers, Tornheim double series and linear Euler sums. Available online at ar**v:1551.03079v3, 2015.

  2. D. H. Bailey, J. M. Borwein, and R. Girgensohn. Experimental evaluation of Euler sums. Exp. Math. 3:1 (1994), 17–30.

    Article  MathSciNet  Google Scholar 

  3. D. H. Bailey, J. M. Borwein, and R. Girgensohn. Finding new mathematical identities with supercomputers. In Proceedings of the Seventh SIAM Conference on Parallel Processing for Scientific Computing. Society for Industrial and Applied Mathematics, 1995.

  4. D. H. Bailey and S. Plouffe. Recognizing numerical constants. In Organic Mathematics Workshop, Canadian Mathematical Society Conference Proceedings, vol. 20. American Mathematical Society, 1997.

  5. D. H. Bailey, J. M. Borwein, V. Kapoor, and E. W. Weisstein. Ten problems in experimental mathematics. Amer. Math. Monthly 113:6 (2006), 481–509.

    Article  MathSciNet  Google Scholar 

  6. D. H. Bailey, J. M. Borwein, N. J. Calkin, R. Girgensohn, D. R. Luke, and V. H. Moll. Experimental Mathematics in Action. A K Peters, 2007.

  7. A. Basu and T. M. Apostol. A new method for investigating Euler sums. Ramanujan J. 4:4 (2000), 397–419.

    Article  MathSciNet  Google Scholar 

  8. J. Boersma. In memoriam P. J. de Doelder. Newsletter of the SIAM Activity Group on Orthogonal Polynomials and Special Functions 5:2 (1994), 3.

  9. D. Borwein and J. M. Borwein. On an intriguing integral and some series related to \(\zeta (4)\). Proc. Amer. Math. Soc. 123:4 (1995), 1191–1198.

    MathSciNet  MATH  Google Scholar 

  10. D. Borwein, J. M. Borwein, and R. Girgensohn. Explicit evaluation of Euler sums. Proc. Edinb. Math. Soc. 38:2 (1995), 277–294.

    Article  MathSciNet  Google Scholar 

  11. J. Borwein and D. Bailey. Mathematics by Experiment: Plausible Reasoning in the 21st Century. A K Peters, 2004.

  12. J. Borwein, D. Bailey, and R. Girgensohn. Experimentation in Mathematics: Computational Paths to Discovery. A K Peters, 2004.

  13. J. M. Borwein and D. M. Bradley. Thirty-two Goldbach variations. Int. J. Number Theory 2:1 (2006), 65–103.

    Article  MathSciNet  Google Scholar 

  14. J. Borwein and K. Devlin. The Computer as Crucible: An Introduction to Experimental Mathematics. A K Peters, 2009.

  15. J. M. Borwein. Exploratory experimentation: Digitally-assisted discovery and proof. In Proof and Proving in Mathematics Education. Springer, 2012.

  16. D. Castellanos. The ubiquitous \(\pi \). Math. Mag. 61:2 (1988), 67–98 (Part I); 61:3 (1988), 148–163 (Part II).

  17. H. Chen. Excursions in Classical Analysis: Pathways to Advanced Problem Solving and Undergraduate Research. Mathematical Association of America, 2010.

  18. J. Choi and H. M. Srivastava. Explicit evaluation of Euler and related sums. Ramanujan J. 10:1 (2005), 51–70.

    Article  MathSciNet  Google Scholar 

  19. W. Chu. Hypergeometric series and the Riemann zeta function. Acta Arith. 82:2 (1997), 103–118.

    Article  MathSciNet  Google Scholar 

  20. P. J. de Doelder. On some series containing \(\psi (x) - \psi (y)\) and \((\psi (x) - \psi (y))^2\) for certain values of \(x\) and \(y\). J. Comput. Appl. Math. 37:1–3 (1991), 125–141.

    Article  MathSciNet  Google Scholar 

  21. R. Dutta. Evaluation of a cubic Euler sum. J. Class. Anal. 9:2 (2016), 151–159.

    Article  MathSciNet  Google Scholar 

  22. L. Euler. De summis serierum reciprocarum. Comm. Acad. Sci. Petropol. 7 (1734), 123–134. Reprinted in Opera Omnia ser. I, vol. 14. B. G. Teubner (1925) 73–86.

  23. L. Euler. Meditationes circa singulare sererum genus. Novi Comm. Acad. Sci. Petropol. 20 (1776), 140–186. Reprinted in Opera Omnia ser. I, vol. 15. B. G. Teubner (1927) 217–267.

  24. H. Eves and E. P. Starke (eds.). The Otto Dunkel Memorial Problem Book. Amer. Math. Monthly 64:7 (Part II) (1957), 45.

  25. P. Flajolet and B. Salvy. Euler sums and contour integral representations. Exp. Math. 7:1 (1998), 15–35.

    Article  MathSciNet  Google Scholar 

  26. P. Freitas. Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums. Math. Comp. 74:251 (2005), 1425–1440.

    Article  MathSciNet  Google Scholar 

  27. O. Furdui. Series involving products of two harmonic numbers. Math. Mag. 84:5 (2011), 371–377.

    Article  MathSciNet  Google Scholar 

  28. O. Furdui. Limits, Series, and Fractional Part Integrals. Springer, 2013.

    Book  Google Scholar 

  29. O. Furdui and A. Sîntămărian. A new proof of the quadratic series of Au-Yeung. Gazeta Matematică Seria A 37:1-2 (2019), 1–6.

    MATH  Google Scholar 

  30. C. Georghiou and A. N. Philippou. Harmonic sums and the zeta function. Fibonacci Quart. 21:1 (1983), 29–36.

    MathSciNet  MATH  Google Scholar 

  31. E. R. Hansen. A Table of Series and Products. Prentice-Hall, 1975.

    MATH  Google Scholar 

  32. R. Harada. On Euler’s formulae for double zeta values. Kyushu J. Math. 72:1 (2018), 15–24.

    Article  MathSciNet  Google Scholar 

  33. S. Kifowit and T. Stamps. The harmonic series diverges again and again. AMATYC Rev. 27:2 (2006), 31–43.

    Google Scholar 

  34. M. Kneser. A summation problem. Solution to Problem 4305. Amer. Math. Monthly 57:4 (1950), 267–268.

  35. L. Lewin. Polylogarithms and Associated Functions. North Holland, 1981.

    MATH  Google Scholar 

  36. P. Mengoli. Novæ Quadraturæ Arithmeticæ seu De Additione Fractionum. Bologna, 1650.

  37. I. Mező. Nonlinear Euler sums. Pacific J. Math. 272:1 (2014), 201–226.

    Article  MathSciNet  Google Scholar 

  38. L. J. Mordell. On the evaluation of some multiple series. J. Lond. Math. Soc. 33:3 (1958), 368–371.

    Article  MathSciNet  Google Scholar 

  39. L. J. Mordell. Note on a harmonic double series. J. Lond. Math. Soc. 35:4 (1960) 401–402.

    Article  MathSciNet  Google Scholar 

  40. S. G. Moreno. A one-sentence and truly elementary proof of the Basel problem. Available online at arxiv.org/abs/1502.07667, 2015.

  41. N. Nielsen. Handbuch der Theorie der Gammafunktion. B. G. Teubner, 1906.

    MATH  Google Scholar 

  42. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark. NIST Handbook of Mathematical Functions. Cambridge University Press, 2010.

    MATH  Google Scholar 

  43. T.-L Rădulescu, V. D. Rădulescu, and T. Andreescu. Problems in Real Analysis: Advanced Calculus on the Real Line. Springer, 2009.

    Book  Google Scholar 

  44. H. F. Sandham. Problem 4305. Amer. Math. Monthly 55:7 (1948), 431.

    Article  MathSciNet  Google Scholar 

  45. L.-C. Shen. Remarks on some integrals and series involving the Stirling numbers and \(\zeta (n)\). Trans. Amer. Math. Soc. 347:4 (1995), 1391–1399.

    MathSciNet  MATH  Google Scholar 

  46. Z. K. Silagadze. The Basel problem: A physicist’s solution. Math. Intelligencer 41:3 (2019), 14–18.

    Article  MathSciNet  Google Scholar 

  47. R. Sita Rama Chandra and A. Siva Rama Sarma. Indian J. Pure Appl. Math. 10:5 (1979), 602–607.

  48. S. M. Stewart. Explicit evaluation of some quadratic Euler-type sums containing double-index harmonic numbers. Tatra Mt. Math. Publ. 77:1 (2020), 73–98.

    MathSciNet  Google Scholar 

  49. L. Tornheim. Harmonic double series. Amer. J. Math. 72:2 (1950), 303–314.

    Article  MathSciNet  Google Scholar 

  50. C. I. Vălean and O. Furdui. Reviving the quadratic series of Au-Yeung. J. Class. Anal. 6:2 (2015), 113–118.

    Article  MathSciNet  Google Scholar 

  51. C. I. Vălean. (Almost) Impossible Integrals, Sums, and Series. Springer, 2019.

    Book  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  53. J. Wolstenholme. A Book of Mathematical Problems, on Subjects Included in the Cambridge Course. Macmillan, 1867.

    Book  Google Scholar 

  54. D. Zagier. The dilogarithm function. In Frontiers in Number Theory, Physics, and Geometry II—On Conformal Field Theories, Discrete Groups and Renormalization. Springer, 2007.

  55. D. Y. Zheng. Further summation formulae related to generalized harmonic numbers. J. Math. Anal. Appl. 335:1 (2007), 692–706.

    Article  MathSciNet  Google Scholar 

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Stewart, S.M. In The Shadow of Euler’s Greatness: Adventures in the Rediscovery of an Intriguing Sum. Math Intelligencer 43, 82–91 (2021). https://doi.org/10.1007/s00283-021-10106-5

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