## Functiones et Approximatio Commentarii Mathematici

### Kondo-Saito-Tanaka theorem

#### Abstract

Given an integer $q\ge 2$, a $q$-normal number is an irrational number $\eta$ such that any preassigned sequence of $k$ digits occurs in the $q$-ary expansion of $\eta$ at the expected frequency, namely $1/q^k$. Given an integer $q\ge 3$, we consider the sequence of primes reduced modulo $q$ and examine various possibilities of constructing normal numbers using this sequence. We create a sequence of independent random variables that mimics the sequence of primes and then show that for almost all outcomes this allows to obtain a normal number.

#### Article information

Source
Funct. Approx. Comment. Math., Volume 49, Number 2 (2013), 291-302.

Dates
First available in Project Euclid: 20 December 2013

https://projecteuclid.org/euclid.facm/1387572233

Digital Object Identifier
doi:10.7169/facm/2013.49.2.8

Mathematical Reviews number (MathSciNet)
MR3127896

Keywords
normal numbers primes

#### Citation

De Koninck, Jean-Marie; Kátai, Imre. Kondo-Saito-Tanaka theorem. Funct. Approx. Comment. Math. 49 (2013), no. 2, 291--302. doi:10.7169/facm/2013.49.2.8. https://projecteuclid.org/euclid.facm/1387572233

#### References

• P.T. Bateman, R.A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comp. 16 (1962), 363–367.
• P.T. Bateman, R.A. Horn, Primes represented by irreducible polynomials in one variable, Proc. Sympos. Pure Math. 8 (1965), 119–135.
• J.M. De Koninck, I. Kátai, Construction of normal numbers by classified prime divisors of integers, Functiones et Approximatio 45.2 (2011), 231–253.
• J.M. De Koninck, I. Kátai, Construction of normal numbers by classified prime divisors of integers II, Functiones et Approximatio 49.1 (2013), 7–27.
• J.M. De Koninck, I. Kátai, Normal numbers created from primes and polynomials, Uniform Distribution Theory 7 (2012), no.2, 1–20.
• J.M. De Koninck, I. Kátai, Some new methods for constructing normal numbers, Annales des Sciences Mathématiques du Québec 36 (2012), no. 2, 349–359.
• H. Halberstam, H.E. Richert, Sieve Methods, Academic Press, New York, 1974.
• J. Galambos, Introductory Probability Theory, Marcel Dekker, New York, 1984.
• A. Schinzel, W. Sierpinski, Sur certaines hypothèses concernant les nombres premiers, Acta Arith. 4 (1958), 185–208; Corrigendum: ibid. 5 (1959), 259.
• D. Shiu, Strings of congruent primes, J. London Math. Soc. (2) 61 (2000), no.2, 359–363.