**Journal of Prime Research in Mathematics**

Vol. 1 (2012), Issue 1, pp. 28 – 35

ISSN: 1817-3462 (Online) 1818-5495 (Print)

ISSN: 1817-3462 (Online) 1818-5495 (Print)

# Gaps in binary expansions of some arithmetic functions and the irrationality of the Euler constant

**Jorge Jimenez Urroz
**Departamento de Matematica Aplicada IV, Universidad Politecnica de Catalunya, Barcelona, Espana.

**Instituto de Matematicas, Universidad Nacional Autonoma de Mexico, Morelia, Michoacan, Mexico.**

Florian Luca

Florian Luca

**Universite Pierre et Marie Curie Paris 6, Institut de Mathematiques de Jussieu, Paris,**

Michel Waldschmidt

Michel Waldschmidt

France.

\(^{1}\)Corresponding Author: jjimenez@ma4.upc.edu

Copyright © 2012

**Jorge Jimenez Urroz , Florian Luca, Michel Waldschmidt**. 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.**Published:**December, 2012.

### Abstract

We show that if \(F_n = 22^n+ 1\) is the nth Fermat number, then the binary digit sum of \(π(F_n)\) tends to infinity with \(n\), where \(π(x)\) is the counting function of the primes \(p ≤ x\). We also show that if \(F_n\) is not prime, then the binary expansion of \(φ(F_n)\) starts with a long string of 1’s, where \(φ\) is the Euler function. We also consider the binary expansion of the counting function of irreducible monic polynomials of degree a given power of 2 over the field \(\mathbb{F}_{2}\). Finally, we relate the problem of the irrationality of Euler constant with the binary expansion of the sum of the divisor function.

#### Keywords:

Binary expansions, prime number theorem, rational approximations to log 2, Fermat numbers, Euler constant, irreducible polynomials over a finite field.