Riemann Sums and Möbius

Abstract

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 ξ ∈ l^r (S), the series ${f_\zeta } = \sum\nolimits_{n \in s} {A\zeta (x)cos2\pi nx}$ converges uniformly to an element of f_ξR₀, 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 ξ_λ 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 f_ξ ∈ Λ_α, α < 1/2, ξ ∈ l^r (S), 0 < r < 2. Using a 1991 estimate by R. C. Baker and G. Harman, one finds GRH implies f_ξ ∈ Λ_α, α < 1/4, ξ ∈ l^r (S), 0 < r < 2. The question of whether R₀ ∩ Λ_α ≠ {0} for some positive α > 0 is open.