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 University Press

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

[1] [1] Baker, A., Transcendental number theory. Cambridge University Press, London-New York, 1975. Google Scholar

[2] [2] Baker, A., Birch, B. J., and Wirsing, E. A., On a problem ofChowla. J. Number Theory 5(1973), 224–236. http://dx.doi.Org/10.1 01 6/0022-314X(73)90048-6 Google Scholar

[3] [3] Davis, H. T., The summation of series. Principia Press of Trinity University, San Antonia, Tex., 1962. Google Scholar

[4] [4] Hurwitz, A., Einige Eigenschaften der Dirichlet Funktionen F(s) = Y.(D/n)n s> diebei der Bestimmung der Klassenzahlen Binârer quadratischer Formen auftreten. Zeitschrift f. Math. u. Physik 27(1882), 86–101. +diebei+der+Bestimmung+der+Klassenzahlen+Binârer+quadratischer+Formen+auftreten.+Zeitschrift+f.+Math.+u.+Physik+27(1882),+86–101.>Google Scholar

[5] [5] Livingston, A. E., The series Y.7=\ f (n) / n for periodic f. Canad. Math. Bull. 8(1965), no. 4, 413–432. http://dx.doi.Org/10.4153/CMB-1965-029-2 Google Scholar

[6] [6] Ram Murty, M., Problems in analytic number theory. Graduate Texts in Mathematics, 206, Readings in Mathematics, Springer, New York, 2008. Google Scholar

[7] [7] Ram Murty, M. and Sardha, N., Euler-Lehmer constants and a conjecture ofErdb's. J. Number Theory 130(2010), no. 12, 2671–2682. http://dx.doi.Org/10.101 6/j.jnt.2O10.07.004 Google Scholar

[8] [8] Ram Murty, M. and Sinha, K., The generalized Dedekind determinant. In: SCHOLAR—a scientific celebration highlighting open lines of arithmetic research, Contemp. Math., 655, American Mathematical Society, Providence, RI, 2015, pp. 153–164. http://dx.doi.Org/!0.1090/conm/655/13232 Google Scholar

Cité par Sources :