Part 6 of 14. The first modular forms one can write down are sums over a lattice. Their coefficients are divisor sums, the first sign that modular forms encode arithmetic.
Part 5 showed that every modular form for SL₂(ℤ) is a polynomial in E₄ and E₆; this part builds them. An Eisenstein series adds up a contribution from every point of a lattice in ℂ, and since the modular group only reshuffles the lattice, the sum comes out modular. Its q-expansion has divisor sums as coefficients, the first sign that modular forms encode arithmetic.
Drag τ below to deform the lattice: τ = i gives a square lattice and τ = e2πi/3 a hexagonal one. Nearby points dominate the sum.
Figure 1. The lattice Λτ = ℤ + τℤ, each point coloured by the size of its term 1/|mτ+n|k; the cross marks the excluded origin. The status line compares the partial sum over the range with the q-expansion.
Near the real axis the lattice flattens and the sum grows; as Im τ grows the sum tends to 1.
For even k ≥ 4 the Eisenstein series of weight k sums over the nonzero points of ℤ + τℤ:
As τ → i∞ only the terms with m = 0 survive, and Σn≠0 n−k = 2ζ(k), so the factor 1/(2ζ(k)) makes the constant term 1. For odd k the terms at mτ + n and −mτ − n cancel.
About R lattice points lie at distance near R, each of size R−k, so the shells add up like Σ R1−k:
At k = 2 the shells add like Σ 1/R. The sum then converges only conditionally, and its value depends on the order of the terms.
Figure 2 sums (mτ + n)−2 at τ = i over growing copies of a shape. Discs and squares tend to 0. Long rectangles lying along the rows tend to 3/π, the value of E₂(i) from its q-series, and upright ones to −3/π; turning a rectangle moves its limit round a circle.
Figure 2. Partial sums of Σ(mτ+n)−2/2ζ(2) at τ = i over the lattice points inside R times the shape in the inset, for R from 2 to 120, drawn in the complex plane. The cross is the limit predicted from the shape's outline; ever longer rectangles approach the dashed circle.
Each shape has a limit, because a thin shell between two copies of the same shape adds ∫e−2iθdθ = 0 in the limit, but different shapes have different limits. This is why E₂ is not modular. Its q-series sums the lattice row by row, and S turns the lattice so that rows become columns: at τ = i that swaps 3/π for −3/π, and in general E₂(−1/τ) = τ2E₂(τ) + 6τ/(πi).
Summing each row first gives the Fourier expansion in q = e2πiτ. The Lipschitz formula, at z = mτ, turns row m into a series in qm:
Row m contributes dk−1 to qn exactly when md = n, so collecting powers of q gives
with ck = −2k/Bk, Bk the Bernoulli number, and σk−1(n) the sum of the (k−1)th powers of the divisors of n.
Figure 3. Row m of the lattice sum is a series in qm whose d-th term carries dk−1. The cells with md = n (orange) all land on qn, and their sum is σk−1(n). Click a cell or use the slider to pick n.
For the first weights,
E₄'s coefficients are all positive and E₆'s after the constant all negative. Both are integers, which the lattice sum does not make obvious, and Part 5 shows E₈ = E₄2 and E₁₀ = E₄E₆.
σk is multiplicative: if gcd(m, n) = 1, then σk(mn) = σk(m)σk(n), because each divisor of mn splits uniquely as d1d2 with d1 | m and d2 | n. At prime powers it is a geometric sum:
In the table, σ3(6) = σ3(2)·σ3(3) = 9·28 = 252.
Figure 4. The first 12 coefficients ck · σk−1(n) of Ek, with the divisors behind each sum. Switching weights shows how fast the coefficients grow with k.
The largest divisor dominates: σk−1(n)/nk−1 = Σd|n d1−k lies between 1 and ζ(k−1), so the coefficients of Ek grow like cknk−1. Weight 2 is the exception again: σ1(n)/n has no ceiling, since at n = m! every d ≤ m divides n and the sum is at least 1 + 1/2 + ··· + 1/m.
Figure 5. σk−1(n)/nk−1 for n up to 3000. Primes (grey) sit just above 1; numbers with many small divisors rise toward ζ(k−1) (dashed). At k = 2 there is no ceiling.
Integrality rests on ck. It is an integer for k = 4, 6, 8, 10 and 14, but B12 = −691/2730 gives c12 = 65520/691, so every q-coefficient of E12 up to n = 1380 has 691 in its denominator. The first exception is the prime 1381 ≡ −1 (mod 691), where 691 divides σ11(n). Exceptions are rare early on, about 1 in 700 up to n = 200 000, but asymptotically almost all n are exceptions. The prime returns in Part 7 as Ramanujan's congruence.
The lattice sum makes modularity clear and the q-expansion the arithmetic; Lipschitz summation connects them, and the same split returns for the discriminant, the j-function and L-functions. Part 7 takes the combination of E₄ and E₆ at weight 12 that vanishes at the cusp, the discriminant Δ.