Lehmer’s nonvanishing conjecture

Multiply power-series factors and collect the number attached to each power. Ramanujan’s product begins q−24q²+252q³…, so its first three coefficients are 1, −24 and 252. Can a later term ever be missing because its coefficient is zero? The full product is q(1−q)²⁴(1−q²)²⁴…; its integer coefficients are called tau values.[1] [2]

The coefficients of a highly structured infinite product can change sign; can one ever vanish exactly? For the weight 12 discriminant form, $\Delta(q)=q\prod_{m\geq1}(1-q^m)^{24}=\sum_{n\geq1}\tau(n)q^n$, require $\tau(n)\ne0$ for every $n\geq1$. If $\prod_{m\geq1}(1-q^m)^{-X}=\sum_{j\geq0}P_j^\sigma(X)q^j$, then $\tau(n)=P_{n-1}^\sigma(-24)$. The index shift comes from the initial factor $q$.[1] [2]

Changing signs in a sequence does not force a zero term. Partial non-root results concern other values in a related polynomial family. What remains missing is exclusion of a zero at every index, not just those checked or covered by special conditions.[1] [2]

The 2026 root-location counterexamples do not establish a zero at −24 or prove its exclusion at every index.[1] [2]

Other powers of the same product can have systematic gaps in their coefficient sequences. Lehmer’s question tests whether the special 24th power avoids every gap. It asks whether the strong arithmetic symmetry of this particular function forces full support of its integer sequence, despite cancellation in the product expansion.[1] [2]

Changing the eta-product exponent connects modular-form coefficients with a polynomial family studied through algebraic and analytic root restrictions. Lehmer singles out one fixed exponent and requires nonvanishing at every coefficient index. Other eta powers can be lacunary, so arithmetic symmetry alone does not imply full support. This question probes whether the discriminant’s additional structure rules out individual missing terms, beyond the weaker fact that its coefficients are not sparse.[1] [2]