Eisenstein series and lattice sums

For τ in a compact subset of the upper half-plane, the real linear map (m,n)↦mτ+n is uniformly invertible. Its length is bounded below by a positive compact-dependent constant times √(m²+n²). The number of integer pairs in a shell of radius roughly R is proportional to R. Thus absolute convergence is controlled by the sum of R to the power 1−k. It converges exactly on the safe range k>2 used here, uniformly on compact subsets.

Normal convergence makes the lattice sum holomorphic and permits reindexing by an invertible integral matrix. The transformed lattice differs by the scalar factor (cτ+d)⁻¹, so homogeneity gives weight k. This proof does not consist of differentiating a divergent series formally. At odd integral k, pairing (m,n) with (−m,−n) gives zero because absolute convergence permits that grouping. At even k≥4 the result has a nonzero limit at infinity, supplied by the lattice row m=0.

For even k≥4, the normalized expansion is Eₖ=1−(2k/Bₖ)Σσₖ₋₁(n)qⁿ, where Bₖ is the Bernoulli number in the convention B₂=1/6 and σᵣ(n) sums the rth powers of positive divisors. The divisor sum arises because an exponent n can be factored as a positive lattice index times a Fourier index. It is arithmetic information extracted from an analytic lattice sum. Individual coefficients can grow polynomially even though the series converges when |q|<1, because exponential q decay dominates that growth.

The two basic examples are E₄=1+240q+2160q²+6720q³+⋯ and E₆=1−504q−16632q²−122976q³+⋯. These are constant-term-one conventions; some texts instead normalize the coefficient of q to one, changing every displayed numerical factor. Products add weights, so combinations of E₄ and E₆ will create further forms. Their existence alone does not yet prove that every full-level form is such a polynomial; the later dimension argument supplies that completeness statement.

At k=2 the absolute lattice sum fails its convergence test, so the previous reindexing proof cannot be recycled unchanged. The holomorphic series E₂=1−24Σσ₁(n)qⁿ exists on the upper half-plane but satisfies an extra term in its transformation law: E₂(γτ)=(cτ+d)²E₂(τ)+6c(cτ+d)/(πi). The anomaly disappears for translations but not for inversion. It is therefore called quasimodular, and it is not a counterexample to the absence of nonzero holomorphic full-level weight-two forms.

Subtracting 3/(πy) gives E₂*, which transforms with pure weight two but is no longer holomorphic because it depends on the imaginary part. The two corrected features cannot be silently combined into an ordinary holomorphic modular form. This distinction matters whenever derivatives or logarithmic derivatives of modular forms appear. Differentiation introduces related anomalous terms, and E₂ can compensate for them in suitable differential operators. Here the exact transformation law is stated as a theorem; the lattice convergence calculation explains why an exceptional phenomenon is possible, rather than proving that formula by itself.