Geodesic
An elementary approach to toy models for D.~H.~Lehmer's conjecture
Izvestiya. Mathematics , Tome 75 (2011) no. 6, pp. 1093-1106.

Voir la notice de l'article provenant de la source Math-Net.Ru

We use the theory of algebraic fields to establish that none of the shells of the lattice $\mathbb{Z}^2$ (resp. $A_2$) is a 4-design (resp. 6-design). We discuss the connection between spherical designs and imaginary quadratic fields.
Keywords: theta functions, spherical $t$-designs, lattices. 
@article{IM2_2011_75_6_a0,
     author = {E. Bannai and Ts. Miezaki and V. A. Yudin},
     title = {An elementary approach to toy models for {D.~H.~Lehmer's} conjecture},
     journal = {Izvestiya. Mathematics },
     pages = {1093--1106},
     publisher = {mathdoc},
     volume = {75},
     number = {6},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2011_75_6_a0/}
}
TY  - JOUR
AU  - E. Bannai
AU  - Ts. Miezaki
AU  - V. A. Yudin
TI  - An elementary approach to toy models for D.~H.~Lehmer's conjecture
JO  - Izvestiya. Mathematics 
PY  - 2011
SP  - 1093
EP  - 1106
VL  - 75
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2011_75_6_a0/
LA  - en
ID  - IM2_2011_75_6_a0
ER  - 
%0 Journal Article
%A E. Bannai
%A Ts. Miezaki
%A V. A. Yudin
%T An elementary approach to toy models for D.~H.~Lehmer's conjecture
%J Izvestiya. Mathematics 
%D 2011
%P 1093-1106
%V 75
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2011_75_6_a0/
%G en
%F IM2_2011_75_6_a0
E. Bannai; Ts. Miezaki; V. A. Yudin. An elementary approach to toy models for D.~H.~Lehmer's conjecture. Izvestiya. Mathematics , Tome 75 (2011) no. 6, pp. 1093-1106. http://geodesic.mathdoc.fr/item/IM2_2011_75_6_a0/

[1] E. Bannai, T. Miezaki, “Toy models for D. H. Lehmer's conjecture”, J. Math. Soc. Japan, 62:3 (2010), 687–705 | DOI | MR | Zbl

[2] P. de la Harpe, C. Pache, “Cubature formulas, geometrical designs, reproducing kernels, and Markov operators”, Infinite groups: geometric, combinatorial and dynamical aspects (Gaeta, Italy, 2003), Progr. Math., 248, Birkhäuser, Basel, 2005, 219–267 | MR | Zbl

[3] P. de la Harpe, C. Pache, B. Venkov, “Construction of spherical cubature formulas using lattices”, St. Petersburg Math. J., 18:1 (2007), 119–139 | DOI | MR | Zbl

[4] C. Pache, “Shells of selfdual lattices viewed as spherical designs”, Internat. J. Algebra Comput., 5:5–6 (2005), 1085–1127 | DOI | MR | Zbl

[5] B. B. Venkov, “On even unimodular extremal lattices”, Proc. Steklov Inst. Math., 165 (1985), 47–52 | MR | Zbl | Zbl

[6] E. Bannai, T. Miezaki, Toy models for D. H. Lehmer's conjecture. II, arXiv: 1004.1520

[7] J. S. Calcut, “Gaussian integers and arctangent identities for $\pi$”, Amer. Math. Monthly, 116:6 (2009), 515–530 | DOI | MR

[8] P. Delsarte, J. M. Goethals, J. J. Seidel, “Spherical codes and designs”, Geometriae Dedicata, 6:3 (1977), 363–388 | DOI | MR | Zbl

[9] D. B. Zagier, Zetafunktionen und quadratische Körper. Eine Einfuhrüng in die höhere Zahlentheorie, Springer-Verlag, Berlin–New York, 1981 | MR | Zbl

[10] D. A. Cox, Primes of the form $x^2+ny^2$. Fermat, class field theory and complex multiplication, Wiley-Intersci. Publ., Wiley, New York, 1989 | MR | Zbl