Conformal designs and D.H. Lehmer's conjecture

Tsuyoshi Miezaki*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)

Abstract

In 1947, Lehmer conjectured that the Ramanujan τ-function τ(m) is non-vanishing for all positive integers m, where τ(m) are the Fourier coefficients of the cusp form δ of weight 12. It is known that Lehmer's conjecture can be reformulated in terms of spherical t-design, by the result of Venkov. In this paper, we show that τ(m) = 0 is equivalent to the fact that the homogeneous space of the moonshine vertex operator algebra (V{music natural sign})m+1 is a conformal 12-design. Therefore, Lehmer's conjecture is now reformulated in terms of conformal t-designs.

Original languageEnglish
Pages (from-to)59-65
Number of pages7
JournalJournal of Algebra
Volume374
DOIs
Publication statusPublished - 2013 Jan 15
Externally publishedYes

Keywords

  • Conformal design
  • Vertex operator algebras

ASJC Scopus subject areas

  • Algebra and Number Theory

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