Journal of Numbers

Volume 2014 (2014), Article ID 437814, 4 pages

http://dx.doi.org/10.1155/2014/437814

Research Article

Complex Roots of Unity and Normal Numbers

Jean Marie De Koninck¹ and Imre Kátai²

¹Département de Mathématiques et de Statistique, Université Laval, Québec, QC, Canada, G1V 0A6

²Computer Algebra Department, Eötvös Loránd University, Pázmány Péter Sétány I/C, Budapest 1117, Hungary

Received 7 December 2013; Accepted 26 May 2014; Published 12 June 2014

Academic Editor: Andrej Dujella

Copyright © 2014 Jean Marie De Koninck and Imre Kátai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Given an arbitrary prime number , set . We use a clever selection of the values of , 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 be the Liouville function (defined by , where ). It is well known that the statement “ as ” is equivalent to the Prime Number Theorem. It is conjectured that if are arbitrary positive integers, then as . This conjecture seems presently out of reach since we cannot even prove that as .

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 , let be the group of complex roots of unity of order . As runs through the primes, let be independent random variables distributed uniformly on . Then, let be defined on by , so that yields a random variable. In their 2011 paper, Indlekofer et al. proved that if stands for a probability space, where () are the independent random variables, then, for almost all , the sequence is a normal sequence over (see Definition 1 below).

Let us now consider a somewhat different setup. Let be a fixed prime number and set . Given an integer , an expression of the form , where each , is called a word of length . We use the symbol to denote the empty word. Then, will stand for the set of words of length over , while will stand for the set of all words over regardless of their length, including the empty word . Similarly, we define to be the set of words over regardless of their length.

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

where for and . To this representation, we associate the word

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

It can be proved using a theorem of Halász (see [2]) that if is defined on the primes by (), then as .

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 and set . Furthermore set and for . Then, consider the sequence of completely multiplicative functions , , defined on the primes by

Then, set

Theorem 2. The sequence is a normal sequence over .

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

Let be a fixed prime and set and . Recall that stands for the group of complex roots of unity of order ; that is,

Let be such that . Moreover, let be a Dirichlet character modulo of order , meaning that the smallest positive integer for which is . (Here stands for the principal character.)

Let . Consider the polynomial

and assume that its degree is at least 1, that is, that there exists one for which . 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 be an infinite set of primes such that for all . For each , let be a character modulo of order . Further set

Then is a normal sequence over .

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

Corollary 4. Let be defined by . Extend the function to by . Let

and consider the -ary expansion of the real number

Then is a normal number in base .

Example 5. Choosing and as the set of primes , then, the number defined by (12) is normal sequence over while defined by (14) is a ternary normal number.

2. Proof of Theorem 2

Let be a fixed positive integer. Let . Recall the notation . Given a positive integer , let be such that . We will now count the number of those 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 .

We then have

Write the polynomial as , so that ; that is, . We then have

where , with .

With integers such that , we now sum both sides of (20) for , and we then obtain that

Setting

it remains to prove that

To prove this, we proceed using standard techniques. Let stand for the number of solutions of the congruence , in which case we have for all primes and integers . Now define the completely multiplicative function implicitly by the relation

thus implying, in light of (5), that

It follows that

Now, observe that since , 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 and let be any word belonging to . Consider the expression

Observe that if is different from . But if , then

Since, for each ,

it follows that

where again stands for the complex conjugate of . Hence, letting stand for the number of occurrences of as a subword in the word , we have

Now 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.

References

K.-H. Indlekofer, I. Kátai, and O. Klesov, “Renewal theorems for some weighted renewal functions,” Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio Computatorica, vol. 34, pp. 179–194, 2011.

G. Halász, “Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen,” Acta Mathematica Academiae Scientiarum Hungaricae, vol. 19, pp. 365–403, 1968.

A. Weil, “On some exponential sums,” Proceedings of the National Academy of Sciences of the United States of America, vol. 34, pp. 204–207, 1948.

H. Iwaniec and E. Kowalski, Analytic Number Theory, vol. 53 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, USA, 2004.