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Trace Formulas Applied to the Riemann ζ-Function

  • Mark S. Ashbaugh
  • Fritz Gesztesy
  • Lotfi Hermi
  • Klaus KirstenEmail author
  • Lance Littlejohn
  • Hagop Tossounian
Chapter
Part of the CRM Series in Mathematical Physics book series (CRM)

Abstract

We use a spectral theory perspective to reconsider properties of the Riemann zeta function. In particular, new integral representations are derived and used to present its value at odd positive integers.

Keywords

Dirichlet Laplacian Trace class operators Trace formulas Riemann zeta function Spectral theory 

Notes

Acknowledgements

We are indebted to the anonymous referee for kindly bringing references [24] and [27] to our attention.

Klaus Kirsten was supported by the Baylor University Summer Sabbatical and Research Leave Program.

References

  1. 1.
    R. Apéry, Irrationalité de ζ(2) et ζ(3). Astérisque 61, 11–13 (1979)MathSciNetzbMATHGoogle Scholar
  2. 2.
    F. Beukers, A note on the irrationality of ζ(2) and ζ(3). Bull. Lond. Math. Soc. 11, 268–272 (1979)CrossRefGoogle Scholar
  3. 3.
    M. Bordag, G.L. Klimchitskaya, U. Mohideen, V.M. Mostepanenko, Advances in the Casimir Effect (Oxford Science Publications, Oxford, 2009)CrossRefGoogle Scholar
  4. 4.
    A.A. Bytsenko, G. Cognola, L. Vanzo, S. Zerbini, Quantum fields and extended objects in space-times with constant curvature spatial section. Phys. Rep. 266, 1–126 (1996)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    S. Coleman, Aspects of Symmetry: Selected Lectures of Sidney Coleman (Cambridge University Press, Cambridge, 1985)CrossRefGoogle Scholar
  6. 6.
    L.A. Dikii, The zeta function of an ordinary differential equation on a finite interval. Izv. Akad. Nauk SSSR Ser. Mat. 19(4), 187–200 (1955)MathSciNetGoogle Scholar
  7. 7.
    L.A. Dikii, Trace formulas for Sturm–Liouville differential operators. Am. Math. Soc. Transl. (2) 18, 81–115 (1961)Google Scholar
  8. 8.
    J.S. Dowker, R. Critchley, Effective Lagrangian and energy momentum tensor in de Sitter space. Phys. Rev. D13, 3224–3232 (1976)ADSGoogle Scholar
  9. 9.
    R.J. Dwilewicz, J. Mináč, Values of the Riemann zeta function at integers. MATerials MATemàtics 2009(6), 26 ppGoogle Scholar
  10. 10.
    E. Elizalde, Ten Physical Applications of Spectral Zeta Functions. Lecture Notes in Physics, vol. 855 (Springer, Berlin, 2012)Google Scholar
  11. 11.
    G. Esposito, A.Y. Kamenshchik, G. Pollifrone, Euclidean Quantum Gravity on Manifolds with Boundary. Fundamental Theories of Physics, vol. 85 (Kluwer, Dordrecht, 1997)Google Scholar
  12. 12.
    S.R. Finch, Mathematical Constants. Encyclopedia of Mathematics and Its Applications, vol. 94 (Cambridge University Press, Cambridge, 2003)Google Scholar
  13. 13.
    F. Gesztesy, K. Kirsten, Effective Computation of Traces, Determinants, and ζ-Functions for Sturm–Liouville Operators. J. Funct. Anal. 276, 520–562 (2019)MathSciNetCrossRefGoogle Scholar
  14. 14.
    I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged edition, prepared by A. Jeffrey, (Academic, San Diego, 1980)Google Scholar
  15. 15.
    M. Haase, The Functional Calculus for Sectorial Operators. Operator Theory: Advances and Applications, vol. 169 (Birkhäuser, Basel, 2006)Google Scholar
  16. 16.
    S.W. Hawking, Zeta function regularization of path integrals in curved space-time. Commun. Math. Phys. 55, 133–148 (1977)ADSCrossRefGoogle Scholar
  17. 17.
    P. Henrici, Applied and Computational Complex Analysis, Vol. I: Power Series–Integration–Conformal Mapping–Location of Zeros, reprinted 1988 (Wiley, New York, 1974)Google Scholar
  18. 18.
    K. Kirsten, Spectral Functions in Mathematics and Physics (Chapman&Hall/CRC, Boca Raton, 2002)Google Scholar
  19. 19.
    K. Kirsten, A.J. McKane, Functional determinants by contour integration methods. Ann. Phys. 308, 502–527 (2003)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    K. Kirsten, A.J. McKane, Functional determinants for general Sturm–Liouville problems. J. Phys. A 37, 4649–4670 (2004)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    M.S. Klamkin (ed.), Problems in Applied Mathematics: Selections from SIAM Review (SIAM, Philadelphia, 1990)zbMATHGoogle Scholar
  22. 22.
    E. Lindelöf, Le Calcul des Résides et ses Applications a la Théorie des Fonctions (Chelsea, New York, 1947)Google Scholar
  23. 23.
    W. Magnus, F. Oberhettinger, R.P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics. Grundlehren, vol. 52, 3rd edn. (Springer, Berlin, 1966)Google Scholar
  24. 24.
    M.S. Milgram, Integral and series representations of Riemann’s zeta function and Dirichlet’s eta function and a medley of related results. J. Math. 2013, article ID 181724, 17p.Google Scholar
  25. 25.
    K.A. Milton, The Casimir Effect: Physical Manifestations of Zero-Point Energy (World Scientific, River Edge, 2001)CrossRefGoogle Scholar
  26. 26.
    D.B. Ray, I.M. Singer, R-Torsion and the Laplacian on Riemannian Manifolds. Adv. Math. 7, 145–210 (1971)MathSciNetCrossRefGoogle Scholar
  27. 27.
    S.K. Sekatskii, Novel integral representations of the Riemann zeta-function and Dirichlet eta-function, closed expressions for Laurent series expansions of powers of trigonometric functions and digamma function, and summation rules, arXiv:1606.02150Google Scholar
  28. 28.
    H.M. Srivastava, J. Choi, Series Associated with the Zeta and Related Functions (Kluwer, Dordrecht, 2001)CrossRefGoogle Scholar
  29. 29.
    A. van der Poorten, A proof that Euler missed …. Apéry’s proof of the irrationality of ζ(3). An informal report. Math. Intell. 1(4), 195–203 (1979)Google Scholar
  30. 30.
    E.T. Whittaker, G.N. Watson, A Course of Modern Analysis, 4th edn., reprinted 1986 (Cambridge University Press, Cambridge, 1927)Google Scholar
  31. 31.
    Wikipedia: Apery’s constant, https://en.wikipedia.org/wiki/Apery's_constant
  32. 32.
    Wikipedia: Riemann zeta function, https://en.wikipedia.org/wiki/Riemann_zeta_function
  33. 33.
    WolframMathWorld: Apery’s constant, http://mathworld.wolfram.com/AperysConstant.html
  34. 34.
    WolframMathWorld: Riemann Zeta Function, http://mathworld.wolfram.com/RiemannZetaFunction.html
  35. 35.
    W. Zudilin, An elementary proof of Apéry’s theorem, arXiv:0202159Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Mark S. Ashbaugh
    • 1
  • Fritz Gesztesy
    • 2
  • Lotfi Hermi
    • 4
  • Klaus Kirsten
    • 5
    Email author
  • Lance Littlejohn
    • 2
  • Hagop Tossounian
    • 2
    • 3
  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA
  2. 2.Department of MathematicsBaylor UniversityWacoUSA
  3. 3.Center for Mathematical ModelingUniversidad de ChileSantiagoChile
  4. 4.Department of Mathematics and StatisticsFlorida International UniversityMiamiUSA
  5. 5.GCAP-CASPER, Department of MathematicsBaylor UniversityWacoUSA

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