TY - JOUR

T1 - Toy models for D. H. Lehmer's conjecture

AU - Bannai, Eiichi

AU - Miezaki, Tsuyoshi

PY - 2010/7

Y1 - 2010/7

N2 - Abstract. In 1947, Lehmer conjectured that the Ramanujan τ-function τ(m) never vanishes for all positive integers m, where τ(m) are the Fourier coefficients of the cusp form Δ24 of weight 12. Lehmer verified the conjecture in 1947 form < 214928639999. In 1973, Serre verified up to m < 1015, and in 1999, Jordan and Kelly for m < 22689242781695999. The theory of spherical t-design, and in particular those which are the shells of Euclidean lattices, is closely related to the theory of modular forms, as first shown by Venkov in 1984. In particular, Ramanujan's τ-function gives the coefficients of a weighted theta series of the E8-lattice. It is shown, by Venkov, de la Harpe, and Pache, that τ(m) = 0 is equivalent to the fact that the shell of norm 2m of the E8-lattice is an 8-design. So, Lehmer's conjecture is reformulated in terms of spherical t-design. Lehmer's conjecture is difficult to prove, and still remains open. In this paper, we consider toy models of Lehmer's conjecture. Namely, we show that the m-th Fourier coefficient of the weighted theta series of the Z2-lattice and the A2-lattice does not vanish, when the shell of normm of those lattices is not the empty set. In other words, the spherical 5 (resp. 7)-design does not exist among the shells in the Z2-lattice (resp. A 2-lattice).

AB - Abstract. In 1947, Lehmer conjectured that the Ramanujan τ-function τ(m) never vanishes for all positive integers m, where τ(m) are the Fourier coefficients of the cusp form Δ24 of weight 12. Lehmer verified the conjecture in 1947 form < 214928639999. In 1973, Serre verified up to m < 1015, and in 1999, Jordan and Kelly for m < 22689242781695999. The theory of spherical t-design, and in particular those which are the shells of Euclidean lattices, is closely related to the theory of modular forms, as first shown by Venkov in 1984. In particular, Ramanujan's τ-function gives the coefficients of a weighted theta series of the E8-lattice. It is shown, by Venkov, de la Harpe, and Pache, that τ(m) = 0 is equivalent to the fact that the shell of norm 2m of the E8-lattice is an 8-design. So, Lehmer's conjecture is reformulated in terms of spherical t-design. Lehmer's conjecture is difficult to prove, and still remains open. In this paper, we consider toy models of Lehmer's conjecture. Namely, we show that the m-th Fourier coefficient of the weighted theta series of the Z2-lattice and the A2-lattice does not vanish, when the shell of normm of those lattices is not the empty set. In other words, the spherical 5 (resp. 7)-design does not exist among the shells in the Z2-lattice (resp. A 2-lattice).

KW - Hecke operator

KW - Lattices

KW - Modular forms

KW - Spherical t-design

KW - Weighted theta series

UR - http://www.scopus.com/inward/record.url?scp=77957958888&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77957958888&partnerID=8YFLogxK

U2 - 10.2969/jmsj/06230687

DO - 10.2969/jmsj/06230687

M3 - Article

AN - SCOPUS:77957958888

VL - 62

SP - 687

EP - 705

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

SN - 0025-5645

IS - 3

ER -