Geodesic
On a Conjecture of Livingston
Canadian mathematical bulletin, Tome 60 (2017) no. 1, pp. 184-195

Voir la notice de l'article provenant de la source Cambridge

DOI

In an attempt to resolve a folklore conjecture of Erdös regarding the non-vanishing at $s\,=\,1$ of the $L$ -series attached to a periodic arithmetical function with period $q$ and values in $\left\{ -1,\,1 \right\}$ , Livingston conjectured the $\overline{\mathbb{Q}}$ -linear independence of logarithms of certain algebraic numbers. In this paper, we disprove Livingston’s conjecture for composite $q\,\ge \,4$ , highlighting that a new approach is required to settle Erdös conjecture. We also prove that the conjecture is true for prime $q\,\ge \,3$ , and indicate that more ingredients will be needed to settle Erdös conjecture for prime $q$ .
DOI : 10.4153/CMB-2016-065-1
Mots-clés : 11J86, 11J72, non-vanishing of L-series, linear independence of logarithms of algebraic numbers 
Pathak, Siddhi. On a Conjecture of Livingston. Canadian mathematical bulletin, Tome 60 (2017) no. 1, pp. 184-195. doi: 10.4153/CMB-2016-065-1
@article{10_4153_CMB_2016_065_1,
     author = {Pathak, Siddhi},
     title = {On a {Conjecture} of {Livingston}},
     journal = {Canadian mathematical bulletin},
     pages = {184--195},
     year = {2017},
     volume = {60},
     number = {1},
     doi = {10.4153/CMB-2016-065-1},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2016-065-1/}
}
TY  - JOUR
AU  - Pathak, Siddhi
TI  - On a Conjecture of Livingston
JO  - Canadian mathematical bulletin
PY  - 2017
SP  - 184
EP  - 195
VL  - 60
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2016-065-1/
DO  - 10.4153/CMB-2016-065-1
ID  - 10_4153_CMB_2016_065_1
ER  - 
%0 Journal Article
%A Pathak, Siddhi
%T On a Conjecture of Livingston
%J Canadian mathematical bulletin
%D 2017
%P 184-195
%V 60
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2016-065-1/
%R 10.4153/CMB-2016-065-1
%F 10_4153_CMB_2016_065_1

Cité par Sources :