Journal d'Analyse Mathématique

, Volume 135, Issue 2, pp 413–436 | Cite as

Riemann Sums and Möbius

  • William A. Veech


Let S be the set of square-free natural numbers. A Hilbert-Schmidt operator, A, associated to the Möbius function has the property that it maps from \({ \cup _{0 < r < \infty }}{l^r}(s)\) to \({ \cap _{0 < r < \infty }}{l^r}(s)\), injectively. If 0 < r< 2 and ξlr (S), the series \({f_\zeta } = \sum\nolimits_{n \in s} {A\zeta (x)cos2\pi nx} \) converges uniformly to an element of fξR0, i.e., a periodic, even, continuous function with equally spaced Riemann sums, \(\sum\nolimits_{j = 0}^{N - 1} {{f_\zeta }} (j/N) = 0,N = 1,2....\) If \({A_{\zeta \lambda }} = \lambda {\zeta _\lambda },{\zeta _\lambda }(1) = 1\), then ξλ is multiplicative. If \({f_{{\zeta _\lambda }}} \in {\Lambda _a}\), the space of α-Lipschitz continous functions, for some α > 0, and if χ is any Dirichlet character, then L(s, χ) ≠ 0, Res > 1 − α. Conjecturally, the Generalized Riemann Hypothesis (GRH) is equivalent to fξ ∈ Λα, α < 1/2, ξlr (S), 0 < r < 2. Using a 1991 estimate by R. C. Baker and G. Harman, one finds GRH implies fξ ∈ Λα, α < 1/4, ξlr (S), 0 < r < 2. The question of whether R0 ∩ Λα ≠ {0} for some positive α > 0 is open.


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  1. [B61]
    A. S. Besicovitch, Problem on continuity, J. London Math. Soc. 36 (1961), 388–392.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [BC63]
    P. T. Bateman and S. Chowla, Some special trigonometrical series related to the distribution of prime numbers, J. London Math. Soc. 38 (1963), 372–374.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [BH91]
    R. C. Baker and G. Harman, Exponential sums formed with the Möbius function, J. London Math. Soc. (2) 43 (1991), 193–198.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [D37]
    H. Davenport, On some infinite series involving arithmetical functions, Quart. J. Math. 8 (1937), 8–13.CrossRefzbMATHGoogle Scholar
  5. [D37-II]
    H. Davenport, On some infinite series involving arithmetical functions (II), Quart. J. Math. 8 (1937), 313–320.CrossRefzbMATHGoogle Scholar
  6. [HS87]
    D. Hajela and B. Smith, On the maximum of an exponential sum of the Möbius function, Number Theory, Springer, Berlin, 1987, pp. 145–164.zbMATHGoogle Scholar
  7. [IK04]
    H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc., Providence, RI, 2004.CrossRefzbMATHGoogle Scholar
  8. [KS63]
    J.-P. Kahane and R. Salem, Ensembles parfaits et séries trigonométriques, Hermann, Paris, 1963.zbMATHGoogle Scholar
  9. [Pri16]
    J. Priwaloff, Sur les fonctions conjuguées, Bull. Soc. Math. France 44 (1916), 100–103.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [Tao10]
    T. Tao, A remark on partial sums involving the Möbius function Bull, Aust. Math. Soc. 81 (2010), 343–349.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [T39]
    E. C. Titchmarsh, The Theory of Functions, reprint of the second (1939) edition, Oxford University Press, Oxford, 1958.Google Scholar
  12. [V86]
    W. A. Veech, Periodic points and invariant pseudomeasures for toral endomorphisms, Ergodic Theory Dyn. Syst. 6 (1986), 449–473.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [Z59]
    A. Zygmund, Trigonometric Series, 2nd ed., Vols. I, II, Cambridge University Press, New York, 1959.zbMATHGoogle Scholar

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© Hebrew University Magnes Press 2018

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  • William A. Veech

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