Abstract
A recent paper of Furdui and Vălean proves some results about sums of products of “tails” of the series for the Riemann zeta function. We show how such results can be proved with weaker hypotheses using multiple zeta values, and also show how they can be generalized to products of three or more such tails.
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References
Arakawa T., Kaneko M.: Multiple zeta values, poly-Bernoulli numbers, and related zeta functions. Nagoya Math. J. 153, 189–209 (1999)
Euler, L: Meditationes circa singulare serierum genus. Novi Comm. Acad. Sci. Petropol. 20, 140–186 (1776). Reprinted in Opera Omnia, Ser. I, vol. 15, pp. 217–267. B. G. Teubner, Berlin (1927)
Freitas P.: Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums. Math. Comput. 74, 1425–1440 (2005)
Furdui, O.: Limits, Series and Fractional Part Integrals: Problems in Mathematical Analysis. Problem Books in Mathematics. Springer, London (2013)
Furdui O., Sîntămărian A.: Quadratic series involving the tail of \({\zeta(k)}\). Integral Transforms Spec. Funct. 26, 1–8 (2015)
Furdui, O., Vălean, C.: Evaluation of series involving the product of the tail of \({\zeta(k)}\) and \({\zeta(k + 1)}\). Mediterr. J. Math. (2014). doi:10.1007/s00009-014-0508-9
Granville, A.: A decomposition of Riemann’s zeta function. In: Motohashi, Y. (ed.) Analytic Number London Mathematical Society Lecture Note Series, vol. 247, pp. 95–101. Cambridge University Press, New York (1997)
Hoffman M.E.: Multiple harmonic series. Pac. J. Math. 152, 275–290 (1992)
Matsumoto, K.: On the analytic continuation of various multiple zeta-functions. In: Bennett, M.A. et al. (eds.) Number Theory for the Millennium, vol. II (Urbana, 2000), pp. 417–440. A. K. Peters, Natick (2002)
Ohno Y.: A generalization of the duality and sum formulas on the multiple zeta values. J. Number Theory 74, 39–43 (1999)
Sloane, N.J.A.: Online encyclopedia of integer sequences. oies.org
Zagier, D.: Values of zeta functions and their applications. In: First European Congress of Mathematics, Vol. II (Paris, 1992), pp. 497–512. Birkhäuser, Boston (1994)
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Hoffman, M.E. Sums of Products of Riemann Zeta Tails. Mediterr. J. Math. 13, 2771–2781 (2016). https://doi.org/10.1007/s00009-015-0653-9
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DOI: https://doi.org/10.1007/s00009-015-0653-9