## Abstract

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 E_{8}-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 E_{8}-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 Z^{2}-lattice and the A_{2}-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 Z_{2}-lattice (resp. A _{2}-lattice).

Original language | English |
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Pages (from-to) | 687-705 |

Number of pages | 19 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 62 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2010 Jul |

Externally published | Yes |

## Keywords

- Hecke operator
- Lattices
- Modular forms
- Spherical t-design
- Weighted theta series

## ASJC Scopus subject areas

- Mathematics(all)