Advertisement

Results in Mathematics

, 74:18 | Cite as

Zeta Identities in Parameter Form

  • Anthony SofoEmail author
Article
  • 22 Downloads

Abstract

We develop new parameterized series representations of zeta functions.

Keywords

Integral forms summation formulas zeta functions parameterized series representations 

Mathematics Subject Classification

11M35 11A67 05A10 

Notes

Acknowledgements

The author is grateful to the referees for their suggestions and careful reading of the paper.

References

  1. 1.
    Alladi, K., Defant, C.: Revisiting the Riemann zeta function at positive even integers. Int. J. Number Theory 14(7), 1849–1856 (2018)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alzer, H., Sondow, J.: A parameterized series representation for Apéry’s constant \(\zeta \left(3\right) \). J. Comput. Anal. Appl. 20(7), 1380–1386 (2016)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Alzer, H., Koumandos, S.: Series and product representations for some mathematical constants. Period. Math. Hungar. 58(1), 71–82 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bailey, D.H., Borwein, D., Borwein, J.M.: On Eulerian log-gamma integrals and Tornheim–Witten zeta functions. Ramanujan J. 36(1–2), 43–68 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Borwein, D., Borwein, J.M., Girgensohn, R.: Explicit evaluation of Euler sums. Proc. Edinburgh Math. Soc. 38(2), 277–294 (1995)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Euler, L.: Opera Omnia, Ser 1, vol. XV, pp. 217–267. Teubner, Berlin (1917)Google Scholar
  7. 7.
    Hasse, H.: Ein Summierungsverfahren für die Riemannsche Zeta-Reihe. Math. Z. 32, 458–464 (1930)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Janous, W.: Around Apéry’s constant. JIPAM J. Inequal. Pure Appl. Math. 7(1), Article 35 (2006)Google Scholar
  9. 9.
    Milgram, M.S.: Integral and series representations of Riemann’s zeta function and Dirichlet’s eta function and a medley of related results. J. Math. 2013, 181724-1–181724-17 (2013)Google Scholar
  10. 10.
    Ribeiro, P.: Another proof of the famous formula for the zeta function at positive even integers. Am. Math. Mon. 125(9), 839–841 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Sofo, A.: Integral identities for sums. Math. Commun. 13(2), 303–309 (2008)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Sofo, A.: Summation formula involving harmonic numbers. Anal. Math. 37(1), 51–64 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Sofo, A.: Quadratic alternating harmonic number sums. J. Number Theory 154, 144–159 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Sondow, J.: Analytic continuation of Riemann’s zeta function and values at negative integers via Euler’s transformation of series. Proc. Am. Math. Soc. 120, 421–424 (1994)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Srivastava, H.M., Choi, J.: Zeta: q-Zeta functions and associated series and integrals. Elsevier Inc., Amsterdam (2012)zbMATHGoogle Scholar
  16. 16.
    Sondow, J., Weisstein, E.W.: Riemann zeta function. From MathWorld–a wolfram web resource. http://mathworld.wolfram.com/RiemannZetaFunction.html. Accessed 1 Mar 2018
  17. 17.
    Yakubovich, S.: Certain identities, connection and explicit formulas for the Bernoulli and Euler numbers and the Riemann zeta-values. Analysis (Berlin) 35(1), 59–71 (2015)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Victoria UniversityMelbourne CityAustralia

Personalised recommendations