Given an arbitrary prime number q, set ξ = e^2πi/q. We use a clever selection of the values of ξ^α, 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.

1. Introduction and Statement of the Results

Let λ(n) be the Liouville function (defined by λ(n)≔(−1) ^Ω(n), where ). It is well known that the statement “∑_n≤x λ(n) = o(x) as x → ∞” is equivalent to the Prime Number Theorem. It is conjectured that if b₁ < b₂ < ⋯<b_k are arbitrary positive integers, then ∑_n≤x λ(n)λ(n + b₁) ⋯ λ(n + b_k) = o(x) as x → ∞. This conjecture seems presently out of reach since we cannot even prove that ∑_n≤x λ(n)λ(n + 1) = o(x) as x → ∞.

The Liouville function belongs to a particular class of multiplicative functions, namely, the class of completely multiplicative functions. Recently, Indlekofer et al. [1] considered a very special function constructed in the following manner. Let ℘ stand for the set of all primes. For each q ∈ ℘, let be the group of complex roots of unity of order q. As p runs through the primes, let ξ_p be independent random variables distributed uniformly on C_q. Then, let be defined on ℘ by f(p) = ξ_p, so that f(n) yields a random variable. In their 2011 paper, Indlekofer et al. proved that if stands for a probability space, where ξ_p (p ∈ ℘) are the independent random variables, then, for almost all ω ∈ Ω, the sequence α = f(1)f(2)f(3)⋯ is a normal sequence over C_q (see Definition 1 below).

Let us now consider a somewhat different setup. Let q ≥ 2 be a fixed prime number and set A_q≔{0,1, …, q − 1}. Given an integer t ≥ 1, an expression of the form i₁i₂ ⋯ i_t, where each i_j ∈ A_q, is called a word of length t. We use the symbol Λ to denote the empty word. Then, will stand for the set of words of length t over A_q, while will stand for the set of all words over A_q regardless of their length, including the empty word Λ. Similarly, we define to be the set of words over C_q regardless of their length.

Given a positive integer n, we write its q-ary expansion as

()

where ε_i(n) ∈ A_q for 0 ≤ i ≤ t and ε_t(n) ≠ 0. To this representation, we associate the word

()

Definition 1. Given a sequence of integers a(1), a(2), a(3), …, one will say that the concatenation of their q-ary digit expansions , denoted by , is a normal sequence if the number is a q-normal number.

It can be proved using a theorem of Halász (see [2]) that if is defined on the primes p by f(p) = ξ_a (a ≠ 0), then ∑_n≤x f(n) = o(x) as x → ∞.

Now, given

, let

. We believe that if

, then

()

If this was true, it would follow that

()

We cannot prove (3), but we can prove the following. Let q ∈ ℘ and set ξ≔e^2πi/q. Furthermore set x_k = 2^k and

for k = 1,2, …. Then, consider the sequence of completely multiplicative functions f_k, k = 1,2, …, defined on the primes p by

()

Then, set

()

Theorem 2. The sequence θ is a normal sequence over C_q.

We now use a famous result of André Weil to construct a large family of normal numbers.

Let q be a fixed prime and set ξ≔e^2πi/q and ξ_a≔e^2πia/q = ξ^a. Recall that C_q stands for the group of complex roots of unity of order q; that is,

()

Let p ∈ ℘ be such that q∣p − 1. Moreover, let χ_p be a Dirichlet character modulo p of order q, meaning that the smallest positive integer t for which is q. (Here χ₀ stands for the principal character.)

Let u₀, u₁, …, u_k−1 ∈ A_q. Consider the polynomial

()

and assume that its degree is at least 1, that is, that there exists one j ∈ {0, …, k − 1} for which u_j ≠ 0. Further set

()

According to a 1948 result of Weil [3],

()

For a proof, see Proposition 12.11 (page 331) in the book of Iwaniec and Kowalski [4].

We can prove the following.

Theorem 3. Let p₁ < p₂ < ⋯ be an infinite set of primes such that q∣p_j − 1 for all . For each , let be a character modulo p_j of order q. Further set

()

Then η is a normal sequence over C_q.

As an immediate consequence of this theorem, we have the following corollary.

Corollary 4. Let φ : C_q → A_q be defined by φ(ξ_a) = a. Extend the function φ to by φ(αβ) = φ(α)φ(β). Let

()

and consider the q-ary expansion of the real number

()

Then κ is a normal number in base q.

Example 5. Choosing q = 3 and {p₁, p₂, p₃, …} = {7,13,19, …} as the set of primes p_j ≡ 1 (mod⁡ 3), then, the number η defined by (12) is normal sequence over {0, e^2πi/3, e^4πi/3} while κ defined by (14) is a ternary normal number.

2. Proof of Theorem 2

Let be a fixed positive integer. Let . Recall the notation ξ = e^2πi/q. Given a positive integer k, let x, y be such that . We will now count the number of those n ∈ [x, x + y] for which () holds.

Consider the polynomial

()

so that in particular

()

Taking the derivatives on both sides of the above equation yields

()

Thus,

()

where

stands for the complex conjugate of z.

We then have

()

Write the polynomial P_d as

, so that

; that is,

. We then have

()

where

, with

With integers x, y such that

, we now sum both sides of (20) for n = x, …, x + y, and we then obtain that

()

Setting

()

it remains to prove that

()

To prove this, we proceed using standard techniques. Let ρ(δ) stand for the number of solutions of the congruence Q(n) ≡ 0 (mod⁡ δ), in which case we have ρ(p^α) = ρ(p) for all primes p > k and integers α ≥ 1. Now define the completely multiplicative function g_k implicitly by the relation

()

thus implying, in light of (5), that

()

It follows that

()

Now, observe that since g_k(p^α) = f_k(p^α) − f_k(p^α−1) = ξ^α−1(ξ − 1), it follows that

()

But, since

, we have

()

Hence, combining (28) with (27) and (26), we obtain (23).

We have thus established that

()

which completes the proof of Theorem 2.

3. Proof of Theorem 3

As we will see, the proof of Theorem 3 is essentially a consequence of Weil’s result (10).

Let

be a fixed positive integer. Fix a prime p and let

be any word belonging to

. Consider the expression

()

Observe that f_β(n) = 0 if

is different from β. But if

, then

()

Since, for each

()

it follows that

()

where again

stands for the complex conjugate of z. Hence, letting M_p(β) stand for the number of occurrences of β as a subword in the word Γ_p, we have

()

Now f_β(n) can be written as

()

where

()

Thus taking into account (8), the Weil inequality (10), and the above relations (34) and (35), we obtain that

()

We have thus shown that

()

thus completing the proof of Theorem 3.

Conflict of Interests

The authors of this paper certify that they have no conflict of interests.

Acknowledgments

Jean Marie De Koninck was supported in part by a grant from NSERC. Imre Kátai was supported by the Hungarian and Vietnamese TET 10-1-2011-0645.

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Complex Roots of Unity and Normal Numbers

Abstract

1. Introduction and Statement of the Results

2. Proof of Theorem 2

3. Proof of Theorem 3

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Complex Roots of Unity and Normal Numbers

Abstract

1. Introduction and Statement of the Results

2. Proof of Theorem 2

3. Proof of Theorem 3

Conflict of Interests

Acknowledgments

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