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.
Florian Luca
Instituto de Matematicas, Universidad Nacional Autonoma de Mexico, Morelia, Michoacan, Mexico.
Michel Waldschmidt
Universite Pierre et Marie Curie Paris 6, Institut de Mathematiques de Jussieu, Paris,
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.