An elementary approach to toy models for D. H. Lehmer's conjecture

TL;DR

This paper provides an elementary, modular form-free proof related to Lehmer's conjecture, connecting spherical design properties of lattice shells with algebraic number theory and recent results by Calcut.

Contribution

It introduces an elementary proof of non-spherical design properties of lattice shells, avoiding modular forms, and explores links with Calcut's results and imaginary quadratic fields.

Findings

Elementary proof for $ au(m) eq 0$ related to spherical 8-designs

Non-spherical design properties of shells of certain lattices

Connections between Calcut's results and imaginary quadratic fields

Abstract

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.