Math Conjectures

Lehmer’s Nonvanishing Conjecture for Ramanujan’s τ\tau-function

OPENMajorConjectureProposed 1947 · Full conjecture

Canonical statement

Define integers τ(n)\tau(n) by
∑n=1∞τ(n)qn=q∏m=1∞(1−qm)24(∣q∣<1). \sum_{n=1}^{\infty}\tau(n)q^n =q\prod_{m=1}^{\infty}(1-q^m)^{24}\qquad(|q|<1).
Then τ(n)≠0\tau(n)\ne0 for every integer n≥1n\ge1.
View source LaTeX
Define integers \(\tau(n)\) by
\[
  \sum_{n=1}^{\infty}\tau(n)q^n
    =q\prod_{m=1}^{\infty}(1-q^m)^{24}\qquad(|q|<1).
\] Then \(\tau(n)\ne0\) for every integer \(n\ge1\).
NT-MOD-019:[Lehmer1947Tau][Serre1981Lacunarity][RouseThorner2019Lehmer]
Number theoryAlgebra and representation theory

Notes

Ramanujan's τ\tau-function records the coefficients of the discriminant cusp form Δ=q∏m≥1(1−qm)24\Delta=q\prod_{m\ge1}(1-q^m)^{24} of weight 1212. In 1947 D. H. Lehmer asked whether τ(n)\tau(n) ever vanishes and conjectured that it does not; he showed that the least nn with τ(n)=0\tau(n)=0, if one exists, must be prime, and verified nonvanishing in an initial range [Lehmer1947Tau].

Because τ\tau is multiplicative and satisfies a recursion at prime powers, a zero anywhere forces a zero at a prime, so the conjecture is equivalent to τ(p)≠0\tau(p)\ne0 for every prime pp. Serre, using Galois representations and the Chebotarev density theorem, showed that the primes with τ(p)=0\tau(p)=0 — should any exist — form a set of density zero, with quantitative estimates [Serre1981Lacunarity]. Rouse and Thorner obtained much stronger density bounds via an explicit form of the Sato–Tate distribution for Δ\Delta [RouseThorner2019Lehmer], and computation has pushed direct verification far beyond Lehmer's original range.

A hypothetical zero is thus confined to an extremely sparse set of primes, yet nothing excludes it entirely. The conjecture remains open; a proof must rule out τ(p)=0\tau(p)=0 for every prime, not merely for almost all.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.