arXiv subjects
Publications and source records attributed to Tsuyoshi Miezaki.
Let $L$ be any integral lattice in the 2-dimensional Euclidean space. Generalizing the earlier works of Hiroshi Maehara and others, we prove that for every integer $n>0$, there is a circle in the plane $\mathbb{R}^{2}$ that passes through exactly $n$ points of $L$.
In the previous paper, we studied the "Toy models for D. H. Lehmer's conjecture". Namely, we showed that the m-th Fourier coefficient of the weighted theta series of the $\mathbb{Z}^2$-lattice and the $A_{2}$-lattice does not vanish, when the shell of norm $m$ of those lattices is not the empty set. In other words, the spherical 4 (resp. 6)-design does not exist among the nonempty shells in the $\mathbb{Z}^2$-lattice (resp. $A_{2}$-lattice). This paper is the sequel to the previous paper. We take 2-dimensional lattices associated to the algebraic integers of imaginary quadratic fields whose class number is either 1 or 2, except for $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$, then, show that the $m$-th Fourier coefficient of the weighted theta series of those lattices does not vanish, when the shell of norm $m$ of those lattices is not the empty set. Equivalently, we show that the corresponding spherical 2-design does not exist among the nonempty shells in those lattices.
In 1947, Lehmer conjectured that the Ramanujan's tau function $τ(m)$ never vanishes for all positive integers $m$, where $τ(m)$ is the $m$-th Fourier coefficient of the cusp form $Δ_{24}$ 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 $τ(m)=0$ 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.
In this paper, we give a new upper bound on the minimum Euclidean weight of Type II $\ZZ_{2k}$-codes and the concept of extremality for the Euclidean weights when $k=3,4,5,6$. Together with the known result, we demonstrate that there is an extremal Type II $\ZZ_{2k}$-code of length $8m$ $(m \le 8)$ when $k=3,4,5,6$.
In 1947, Lehmer conjectured that the Ramanujan $τ$-function $τ(m)$ never vanishes for all positive integers $m$, where the $τ(m)$ are the Fourier coefficients of the cusp form $Δ_{24}$ of weight 12. Lehmer verified the conjecture in 1947 for $m<214928639999$. In 1973, Serre verified up to $m<10^{15}$, 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 $\mathbb{Z}^2$-lattice and the $A_{2}$-lattice does not vanish, when the shell of norm $m$ 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 $\mathbb{Z}^2$-lattice (resp. $A_{2}$-lattice).
We locate all of the zeros of the Eisenstein series associated with the Fricke groups $Γ_0^{*}(2)$ and $Γ_0^{*}(3)$ in their fundamental domains by applying and expanding the method of F. K. C. Rankin and H. P. F. Swinnerton-Dyer (``{\it On the zeros of Eisenstein series}'', 1970).