Complex Roots of Unity and Normal Numbers

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