The lacunarity of the powers of the Dedekind eta-function: the exceptional exponent 26, odd exponents, and lacunarity modulo m

Serre (1985) proved that the power $\eta^r$ of the Dedekind eta-function is lacunary — supported on a set of density zero — if and only if $r \in \{2,4,6,8,10,14,26\}$ among positive even integers $r$, and that for even $r$ the coefficients are lacunary modulo every integer $m$. His paper leaves four precise questions open, all still unresolved as of 2026. (1) Are the two known sufficient conditions for $p_{26}(n) = 0$ — $13+12n$ divisible to odd exponent both by a prime $p \equiv -1 \pmod 4$ and by a prime $p' \equiv -1 \pmod 3$, or $13+12n$ a square with all prime factors $\equiv -1 \pmod{12}$ — also necessary? (2) Does $M_{26}(x)$, the number of $n \le x$ with $p_{26}(n) \ne 0$, admit the asymptotic $c\,x/\sqrt{\log x}$ with an explicit constant, as the analogous counting functions do for $r \in \{4,6,8,10,14\}$? (3) Is $\eta^r$ non-lacunary for every odd $r \ge 5$ (conjecturally $M_r(x) \sim x$)? (4) Does lacunarity modulo $m$ extend to odd exponents — e.g., is $\eta^5$ lacunary mod 3, or $\eta^7$ lacunary mod 5? The later literature supplies an explicit formula for $p_{26}(n)$ with a converse in two arithmetic progressions, extensive numerics on odd powers, and lacunarity criteria for integer-weight eta-quotients, but no resolution of any component. The problem is to close these four gaps, completing the lacunarity classification of the integer powers of $\eta$ over $\mathbb{C}$ and modulo $m$.

Background

The Dedekind eta-function is $\eta(z) = q^{1/24}\prod_{m \ge 1}(1 - q^m)$ with $q = e^{2\pi i z}$, defined on the complex upper half-plane. For an integer $r \ge 1$, write $\prod_{m \ge 1}(1-q^m)^r = \sum_{n \ge 0} p_r(n)\, q^n$; for $r = 24$ the coefficients are (up to a shift of index) Ramanujan's tau-function (the Fourier coefficients of the discriminant cusp form $\Delta$, the unique normalized cusp form of weight 12 on $\mathrm{SL}_2(\mathbb{Z})$), and for $r = 1$ Euler's pentagonal number theorem shows $p_1(n)$ vanishes off the generalized pentagonal numbers $(3k^2 \pm k)/2$, $k \in \mathbb{Z}$. A power series is called lacunary if its set of nonzero coefficients has natural density zero; for $\eta^r$ this is measured by $M_r(x) = \#\{n \le x : p_r(n) \ne 0\}$, and lacunarity means $M_r(x) = o(x)$.

For even $r$, $\eta^r$ is a cusp form of integer weight $r/2$ — a modular form, i.e., a holomorphic function on the upper half-plane transforming by a prescribed automorphy factor under a congruence subgroup of the modular group $\mathrm{SL}_2(\mathbb{Z})$, vanishing at the cusps (the boundary points of the modular domain); the weight is the exponent governing the automorphy factor. Serre proved the dichotomy: $\eta^r$ is lacunary exactly for $r \in \{2,4,6,8,10,14,26\}$. The mechanism is his lacunarity criterion: an integer-weight cusp form is lacunary if and only if it is a linear combination of CM forms (CM abbreviates complex multiplication; here, forms built from imaginary quadratic fields), i.e., of theta series attached to Hecke characters of imaginary quadratic fields — a Hecke character being an algebraic character of the ideal group of such a field, and the attached theta series the $q$-series $\sum_{\mathfrak{a}} \chi(\mathfrak{a})\, q^{N(\mathfrak{a})}$ over the integral ideals $\mathfrak{a}$, with $N$ the norm. For the eta powers the relevant fields are $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$, whose theta series have nonzero coefficients only at integers represented by the corresponding norm forms (sums of two squares, and the Eisenstein form $a^2 + ab + b^2$, respectively); this restricted support is what forces the density-zero coefficient support of the lacunary eta powers. The 'only if' direction uses the $\ell$-adic Galois representations attached to Hecke eigenforms (a Hecke eigenform is a simultaneous eigenvector for the Hecke operators, a commuting family of linear operators on modular-form spaces; the attached Galois representation is a continuous linear action of the absolute Galois group of $\mathbb{Q}$ on an $\ell$-adic vector space encoding the coefficients) together with the Chebotarev density theorem, which equidistributes rational primes among the conjugacy classes of that Galois group.

The exponent $r = 26$ is exceptional: $\eta^{26}(12z) = \sum_n p_{26}(n) q^{12n+13}$ has weight 13 and splits into a combination of four CM newforms (a newform is a normalized Hecke eigenform). This yields two explicit sufficient conditions for vanishing: (i) $13 + 12n$ is divisible to odd exponent both by some prime $p \equiv -1 \pmod 4$ and by some prime $p' \equiv -1 \pmod 3$; (ii) $13 + 12n$ is a perfect square, all of whose prime factors are $\equiv -1 \pmod{12}$. Whether these are the only vanishing cases is unknown (Serre checked $n \le 1500$). Serre also proved sharp counting laws for the lacunary even exponents: $M_2(x) \sim c_2\, x/(\log x)^{3/4}$ with $c_2 = 2.4185\ldots$, and $M_r(x) \sim c_r\, x/(\log x)^{1/2}$ for $r \in \{4,6,8,10,14\}$ — Landau–Ramanujan-type asymptotics (counting integers whose prime factorizations have restricted exponent patterns, as in the classical theorem that the integers representable as a sum of two squares have counting function $\sim C\,x/\sqrt{\log x}$). For $r = 26$ only two-sided bounds $a\,x/\sqrt{\log x} \le M_{26}(x) \le b\,x/\sqrt{\log x}$ with $0 < a < b$ are known, and the asymptotic, with its constant, remains open.

For odd $r \ge 5$, $\eta^r$ is a cusp form of half-integral weight $r/2$ (the modular-form framework extends to weights in $\tfrac{1}{2}\mathbb{Z}$), where no lacunarity criterion is known: the Galois-representation argument does not apply directly. Infinitely many zeros do exist — $p_5(n) = 0$ for many $n$, $p_7(28017) = 0$, $p_{15}(53) = 0$, and each single zero propagates to infinitely many via the Hecke operators $T(p^2)$ (the half-integral-weight members of the Hecke family, indexed by squares of primes, which relate Fourier coefficients at indices differing by a square factor) acting on half-integral-weight eigenforms — and Serre conjectures that nonetheless $\eta^r$ is not lacunary, in the strong form $M_r(x) \sim x$. A parallel notion is lacunarity modulo $m$: $\eta^r$ is lacunary mod $m$ if $p_r(n) \not\equiv 0 \pmod m$ only on a set of density zero. Serre proved this for every $m$ when $r$ is even, and for odd $r$ it is known for $m = 2$ (via the congruence $p_r(n) \equiv p_{2r}(2n) \pmod 2$, which reduces to even weight) and for $\eta^5$ mod 5, $\eta^{15}$ mod 5, $\eta^7$ mod 7 (via Freshman's-dream congruences, i.e., binomial-theorem type identities $(1-x)^p \equiv 1 - x^p \pmod p$, yielding for instance $p_5(n) \equiv p_1(n/5) \pmod 5$); the cases $\eta^5$ mod 3 and $\eta^7$ mod 5 are explicitly open.

Four decades of subsequent work — explicit formulas for $p_{26}(n)$, large-scale computations, lacunarity criteria for integer-weight eta-quotients, and parity results — has addressed pieces of this landscape without resolving any of the four questions. Together they constitute the remaining gaps in the complete determination of the lacunarity behavior of the integer powers of $\eta$, over $\mathbb{C}$ and modulo $m$.

Problem Statement

The common objective is to complete the lacunarity classification of the integer powers $\eta^r$ of the Dedekind eta-function — over $\mathbb{C}$ and modulo integers $m$. Serre's theorems settle the even exponents except for two gaps at $r = 26$, and settle the odd exponents modulo no odd modulus beyond a few sporadic cases. The four components that follow are independently well-posed: each may be resolved separately.

(1) Zeros of $p_{26}$. Decide whether the two known sufficient conditions are also necessary: prove that $p_{26}(n) = 0$ implies that $13 + 12n$ is divisible to odd exponent both by some prime $p \equiv -1 \pmod 4$ and by some prime $p' \equiv -1 \pmod 3$, or that $13 + 12n$ is a perfect square all of whose prime divisors are $\equiv -1 \pmod{12}$; or else exhibit an integer $n \ge 0$ with $p_{26}(n) = 0$ satisfying neither condition, or give a complete classification of the additional vanishing families.

(2) Asymptotic for $M_{26}$. Prove or disprove that there exists a constant $c_{26} > 0$ with $M_{26}(x) = \#\{n \le x : p_{26}(n) \ne 0\} \sim c_{26}\, x/\sqrt{\log x}$; in the affirmative case, determine $c_{26}$ explicitly (e.g., as products of Euler factors over residue classes of primes, as Serre did for $r = 2, 4, 6, 8, 10, 14$).

(3) Odd exponents. Determine whether any odd $r \ge 5$ makes $\eta^r$ lacunary. The conjectural answer is that none does: proving that no odd $r \ge 5$ is lacunary resolves this component, while the stronger form $M_r(x) \sim x$ for every odd $r \ge 5$ is the conjectured sharp target. A refutation is an exhibited lacunary odd power.

(4) Lacunarity modulo m for odd exponents. Determine for which pairs $(r, m)$ with $r \ge 5$ odd and $m \ge 2$ the power $\eta^r$ is lacunary modulo $m$; a complete resolution of this component requires such a determination for every such pair, including $\eta^5$ mod 3 and $\eta^7$ mod 5.

Relationship among the components: component (2) would be expected to follow from a positive resolution of (1) by standard Landau-type counting of the multiplicative sets the two conditions define, but the two are not equivalent, since (2) with unspecified constants does not settle (1). Components (3) and (4) are the half-integral-weight counterparts of the classification; component (4) for $m = 2$ is already known.

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