  In 1947, Lehmer conjectured that the Ramanujan's tau function $\tau (m)$
never vanishes for all positive integers $m$, where $\tau (m)$ is the $m$-th
Fourier coefficient of the cusp form $\Delta_{24}$ of weight 12. The theory of
spherical $t$-design is closely related to Lehmer's conjecture because it is
shown, by Venkov, de la Harpe, and Pache, that $\tau (m)=0$ is equivalent to
the fact that the shell of norm $2m$ of the $E_{8}$-lattice is a spherical
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. However,
Bannai-Miezaki showed that none of the nonempty shells of the integer lattice
$\ZZ^2$ in $\RR^2$ is a spherical 4-design, and that none of the nonempty
shells of the hexagonal lattice $A_2$ is a spherical 6-design. Moreover, none
of the nonempty shells of the integer lattices associated to the algebraic
integers of imaginary quadratic fields whose class number is either 1 or 2,
except for $\QQ(\sqrt{-1})$ and $\QQ(\sqrt{-3})$ is a spherical 2-design. In
the proof, the theory of modular forms played an important role.
  Recently, Yudin found an elementary proof for the case of $\ZZ^{2}$-lattice
which does not use the theory of modular forms but uses the recent results of
Calcut. In this paper, we give the elementary (i.e., modular form free) proof
and discuss the relation between Calcut's results and the theory of imaginary
quadratic fields.
