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On systolic zeta functions

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Abstract

We define Dirichlet type series associated with homology length spectra of Riemannian, or Finsler, manifolds, or polyhedra, and investigate some of their analytical properties. As a consequence we obtain an inequality analogous to Gromov’s classical intersystolic inequality, but taking the whole homology length spectrum into account rather than just the systole.

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References

  1. Apostol, T.M.: Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics. Springer, New York (1976)

    Google Scholar 

  2. Babenko, I.K.: Asymptotic volume of tori and the geometry of convex bodies. Math. Notes 44(2), 579–586 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. Babenko, I.K.: Asymptotic invariants of smooth manifolds. Russian Acad. Sci. Izv. Math. 41(1), 1–38 (1992)

    MathSciNet  MATH  Google Scholar 

  4. Babenko, I.K.: Forte souplesse intersystolique de variétés fermées et de polyèdres. Ann. Inst. Fourier (Grenoble) 52(4), 1259–1284 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Babenko, I., Balacheff, F.: Sur la forme de la boule unité de la norme stable unidimensionnelle. Manuscripta Math. 119(3), 347–358 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Burago, D.Yu.: Periodic metrics. In: Vershik, A.M. (ed.) Representation Theory and Dynamical Systems. Advances in Soviet Mathematics, vol. 9, pp. 205–210. American Mathematical Society, Providence (1992)

  7. Cerocchi, F., Sambusetti, A.: A quantitative bounded distance theorem and a Margulis’ lemma for \({\mathbb{Z}}^n\)-actions, with applications to homology. Groups Geom. Dyn. 10(4), 1227–1247 (2016)

  8. Ehrhart, E.: Sur les polyèdres rationnels homothétiques à \(n\) dimensions. C. R. Acad. Sci. Paris 254, 616–618 (1962)

    MathSciNet  MATH  Google Scholar 

  9. Federer, H.: Geometric Measure Theory. Die Grundlehren der Mathematischen Wissenschaften, vol. 153. Springer, New York (1969)

  10. Federer, H.: Real flat chains, cochains and variational problems. Indiana Univ. Math. J. 24(4), 351–407 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gromov, M.: Filling Riemannian manifolds. J. Differential Geom. 18(1), 1–147 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces. Modern Birkhäuser Classics. Birkhaüser, Boston (2007)

    Google Scholar 

  13. Hardy, G.H., Riesz, M.: The General Theory of Dirichlet’s Series. Cambridge University Press, Cambridge (1915)

    MATH  Google Scholar 

  14. Kashin, B.S.: Parallelepipeds of least volume that contain a convex body. Mat. Zametki 45(2), 134–135 (1989) (in Russian)

  15. Massart, D.: Normes Stables des Surfaces. Ph.D. thesis, Ecole Normale Supérieure de Lyon (1996)

  16. Massart, D.: Stable norms of surfaces: local structure of the unit ball of rational directions. Geom. Funct. Anal. 7(6), 996–1010 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Massart, D., Parlier, H.: On the homology length spectrum of surfaces. Int. Math. Res. Not. IMRN 2017(8), 2367–2401 (2017)

    MathSciNet  Google Scholar 

  18. Milnor, J.: Eigenvalues of the Laplace operator on certain manifolds. Proc. Natl. Acad. Sci. USA 51(4), 542 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pełczyński, A., Szarek, S.J.: On parallelepipeds of minimal volume containing a convex symmetric body in \({\mathbb{R}}^n\). Math. Proc. Cambridge Philos. Soc. 109(1), 125–148 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  20. Serre, J.-P.: Cours d’Arithmétique. Le Mathématicien, vol. 2. Presses Universitaires de France, Paris (1970)

    Google Scholar 

  21. Shchepin, E.: Greedy sums and Dirichlet series (2011). arXiv:1110.5285v1

  22. Tenenbaum, G.: Introduction à la Théorie Analytique et Probabiliste des Nombres. Cours Spécialisé, vol. 1. Société Mathématique de France, Paris (1995)

  23. Witt, E.: Eine Identität zwischen Modulformen zweiten Grades. Abh. Math. Semin. Univ. Hambg 14, 323–337 (1941)

    Article  MATH  Google Scholar 

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Correspondence to Daniel Massart.

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Partially supported by the Grants RFSF 10-01-00257-a, and ANR Finsler.

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Babenko, I., Massart, D. On systolic zeta functions. European Journal of Mathematics 3, 899–915 (2017). https://doi.org/10.1007/s40879-017-0181-1

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  • DOI: https://doi.org/10.1007/s40879-017-0181-1

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