• Si, Xin (School of Applied Mathematics Xiamen University of Technology) ;
  • Xu, Ce (School of Mathematical Sciences Xiamen University)
  • Received : 2019.04.04
  • Accepted : 2019.07.04
  • Published : 2020.03.31


This paper develops an approach to the evaluation of quadratic Euler sums that involve harmonic numbers. The approach is based on simple integral computations of polylogarithms. By using the approach, we establish some relations between quadratic Euler sums and linear sums. Furthermore, we obtain some closed form representations of quadratic sums in terms of zeta values and linear sums. The given representations are new.


Supported by : National Natural Science Foundation of China, Natural Science Foundation of Fujian Province


  1. G. E. Andrews, R. Askey, and R. Roy, Special functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, Cambridge, 1999.
  2. D. H. Bailey, J. M. Borwein, and R. E. Crandall, Computation and theory of extended Mordell-Tornheim-Witten sums, Math. Comp. 83 (2014), no. 288, 1795-1821.
  3. D. H. Bailey, J. M. Borwein, and R. Girgensohn, Experimental evaluation of Euler sums, Experiment. Math. 3 (1994), no. 1, 17-30.
  4. B. C. Berndt, Ramanujan's Notebooks. Part I, Springer-Verlag, New York, 1985.
  5. B. C. Berndt, Ramanujan's Notebooks. Part II, Springer-Verlag, New York, 1989.
  6. D. Borwein, J. M. Borwein, and R. Girgensohn, Explicit evaluation of Euler sums, Proc. Edinburgh Math. Soc. (2) 38 (1995), no. 2, 277-294.
  7. J. Borwein, P. Borwein, R. Girgensohn, and S. Parnes, Making sense of experimental mathematics, Math. Intelligencer 18 (1996), no. 4, 12-18.
  8. J. M. Borwein, D. M. Bradley, D. J. Broadhurst, and Petr. Lisonek, Special values of multiple polylogarithms, Trans. Amer. Math. Soc. 353 (2001), no. 3, 907-941. https: //
  9. J. M. Borwein and R. Girgensohn, Evaluation of triple Euler sums, Electron. J. Combin. 3 (1996), no. 1, Research Paper 23, approx. 27 pp.
  10. J. M. Borwein, I. J. Zucker, and J. Boersma, The evaluation of character Euler double sums, Ramanujan J. 15 (2008), no. 3, 377-405.
  11. L. Comtet, Advanced Combinatorics, revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974.
  12. M. Eie and C.-S. Wei, Evaluations of some quadruple Euler sums of even weight, Funct. Approx. Comment. Math. 46 (2012), part 1, 63-77.
  13. P. Flajolet and B. Salvy, Euler sums and contour integral representations, Experiment. Math. 7 (1998), no. 1, 15-35.
  14. P. Freitas, Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums, Math. Comp. 74 (2005), no. 251, 1425-1440.
  15. C. Markett, Triple sums and the Riemann zeta function, J. Number Theory 48 (1994), no. 2, 113-132.
  16. I. Mezo, Nonlinear Euler sums, Pacific J. Math. 272 (2014), no. 1, 201-226.
  17. A. Sofo, Quadratic alternating harmonic number sums, J. Number Theory 154 (2015), 144-159.
  18. P. Sun, The sixth-order sums of the Riemann zeta function, Acta Math. Sinica (Chin. Ser.) 50 (2007), no. 2, 373-384.
  19. C. Xu and J. Cheng, Some results on Euler sums, Funct. Approx. Comment. Math. 54 (2016), no. 1, 25-37.
  20. C. Xu, Y. Yan, and Z. Shi, Euler sums and integrals of polylogarithm functions, J. Number Theory 165 (2016), 84-108.
  21. C. Xu, Y. Yang, and J. Zhang, Explicit evaluation of quadratic Euler sums, Int. J. Number Theory 13 (2017), no. 3, 655-672.