Let S be the set of square-free natural numbers. A Hilbert-Schmidt operator, A, associated to the Mobius 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 ∈ lr (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 0 is open.