Geodesic
The logarithm of irrational numbers and Beatty sequences
Geremias Polanco1
1 School of Natural Science Hampshire College 893 West St. Amherst, MA 01002, U.S.A. and Facultad de Ciencias Instituto de Matemáticas Universidad Autónoma de Santo Domingo Av. Alma Mater Santo Domingo 10105, Dominican Republic
Acta Arithmetica, Tome 179 (2017) no. 2, pp. 101-123 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

Voir la notice de l'article

We find an identity that expresses the logarithm of the ratio of any two irrational numbers $a, b \gt 1$ via a series whose terms are ratios of elements of the Beatty sequences generated by $a$ and $b$. We also show that Sturmian sequences can be defined in terms of these ratios. Furthermore, we find an identity for such series that bears a superficial resemblance to (a discrete version of) Frullani’s integral.
DOI : 10.4064/aa6695-1-2017
Keywords: identity expresses logarithm ratio irrational numbers via series whose terms ratios elements beatty sequences generated sturmian sequences defined terms these ratios furthermore identity series bears superficial resemblance discrete version frullani integral

Geremias Polanco 1

1 School of Natural Science Hampshire College 893 West St. Amherst, MA 01002, U.S.A. and Facultad de Ciencias Instituto de Matemáticas Universidad Autónoma de Santo Domingo Av. Alma Mater Santo Domingo 10105, Dominican Republic 
@article{10_4064_aa6695_1_2017,
     author = {Geremias Polanco},
     title = {The logarithm of irrational numbers and {Beatty} sequences},
     journal = {Acta Arithmetica},
     pages = {101--123},
     year = {2017},
     volume = {179},
     number = {2},
     doi = {10.4064/aa6695-1-2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4064/aa6695-1-2017/}
}
TY  - JOUR
AU  - Geremias Polanco
TI  - The logarithm of irrational numbers and Beatty sequences
JO  - Acta Arithmetica
PY  - 2017
SP  - 101
EP  - 123
VL  - 179
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.4064/aa6695-1-2017/
DO  - 10.4064/aa6695-1-2017
LA  - en
ID  - 10_4064_aa6695_1_2017
ER  - 
%0 Journal Article
%A Geremias Polanco
%T The logarithm of irrational numbers and Beatty sequences
%J Acta Arithmetica
%D 2017
%P 101-123
%V 179
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4064/aa6695-1-2017/
%R 10.4064/aa6695-1-2017
%G en
%F 10_4064_aa6695_1_2017
Geremias Polanco. The logarithm of irrational numbers and Beatty sequences. Acta Arithmetica, Tome 179 (2017) no. 2, pp. 101-123. doi: 10.4064/aa6695-1-2017

Cité par Sources :