Abstract
Given an arbitrary prime number q, set ξ=e2πi/q. We use a clever selection of the values of ξα, α=1,2,…, in order to create normal numbers. We also use a famous result of André Weil concerning Dirichlet characters to construct a family of normal numbers.
Highlights
Introduction and Statement of the ResultsLet λ(n) be the Liouville function (defined by λ(n) := (−1)Ω(n), where Ω(n) := ∑pα‖n α)
The Liouville function belongs to a particular class of multiplicative functions, namely, the class M∗ of completely multiplicative functions
Indlekofer et al [1] considered a very special function f ∈ M∗ constructed in the following manner
Summary
Let λ(n) be the Liouville function (defined by λ(n) := (−1)Ω(n), where Ω(n) := ∑pα‖n α). A(1) a(2) a(3) ⋅ ⋅ ⋅ is a q-normal number It can be proved using a theorem of Halasz (see [2]) that if f ∈ M∗ is defined on the primes p by f(p) = ξa (a ≠ 0), ∑n≤x f(n) = o(x) as x → ∞. Let q be a fixed Recall that Cq prime and set ξ := e2πi/q and ξa := stands for the group of complex e2πia/q = roots of unity of order q; that is, Cq = {ς ∈ C : ςq = 1} = {ξa : a = 0, 1, . . .} as the set of primes pj ≡ 1 (mod 3), the number η defined by (12) is normal sequence over {0, e2πi/3, e4πi/3}. While κ defined by (14) is a ternary normal number
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