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, so this part builds them and their relatives. The Eisenstein series are the simplest modular forms. Each one 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. Their q-expansions are also where the arithmetic first appears, since the coefficients are divisor sums.
Drag τ below to deform the lattice. Nearby points dominate the sum and distant ones fade. τ = i gives a square lattice and τ = e2πi/3 a hexagonal one.
The lattice Λτ = ℤ + τℤ, each point coloured by the size of its term 1/|mτ+n|k. The cross marks the excluded origin. Raising the range shows the partial sum converging.
Near the real axis the lattice flattens and the sum grows; as Im τ grows it stretches and the sum tends to 1. The status bar compares the slow lattice sum with the q-expansion.
Fix an even integer k ≥ 4. The Eisenstein series of weight k is defined by summing over all nonzero points of the lattice ℤ + τℤ:
The sum runs over (m, n) ∈ ℤ² \ {(0, 0)}. The factor 1/(2ζ(k)) fixes the constant term at 1. As τ → i∞ only the terms with m = 0 survive, and Σn≠0 n−k = 2ζ(k) for even k.
The sum converges absolutely for k ≥ 4 but not for k = 2. The lattice has ~R points at radius R, each contributing 1/Rk, so the annulus contribution is R1−k, summable iff k > 2. At k = 2 the sum does not converge absolutely; summed in a fixed order it converges, but to a function that is only quasi-modular.
The row m = 0 alone would converge for k ≥ 2; it is the two-dimensional count of points that pushes the bound to k ≥ 4. Because SL₂(ℤ) permutes the lattice points, Ek is modular, and it is holomorphic on ℍ and bounded at the cusp.
For odd k the series vanishes: terms (mτ + n)−k and (−mτ − n)−k = −(mτ + n)−k cancel.
For computation the Fourier expansion in q = e2πiτ is better. Splitting the double sum row by row and applying the Lipschitz summation formula gives
with ck = −2k/Bk, Bk the k-th Bernoulli number, and σk−1(n) the divisor sum defined below. For the main weights:
E4 is the nonconstant modular form of lowest weight: M2 = 0 and dim M4 = 1, so every weight-4 form is a scalar multiple of E4.
The coefficient of qn is 240 σ3(n). Since σ3(1) = 1, σ3(2) = 1 + 8 = 9, σ3(3) = 1 + 27 = 28, σ3(4) = 1 + 8 + 64 = 73, we get:
M6 is one-dimensional, spanned by E6, and by Part 5 every modular form is a polynomial in E4 and E6.
Now the coefficient of qn is −504 σ5(n). These grow quickly: σ5(2) = 1 + 32 = 33, so the second coefficient is −504 · 33 = −16 632.
E4's coefficients are all positive, and every coefficient of E6 after the constant term is negative. Both have integer coefficients, which the lattice-sum definition does not make obvious. The next two are products of these: Part 5 shows E8 = E42 and E10 = E4E6.
The divisor sum function σk(n) is the sum of the k-th powers of all positive divisors of n:
For instance, the divisors of 12 are {1, 2, 3, 4, 6, 12}, so σ3(12) = 1 + 8 + 27 + 64 + 216 + 1728 = 2044. σ0(n) counts divisors and σ1(n) adds them up.
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. So it is enough to know σk at prime powers, where it is a geometric sum:
At a prime, σk(p) = 1 + pk, so the coefficient of qp in Ek is ck(1 + pk−1) and grows like pk−1. In the table below, σ3(6) = σ3(2) · σ3(3) = 9 · 28 = 252.
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.
Since σk−1(n) is always an integer, integrality rests on ck = −2k/Bk. It is an integer for k = 4, 6, 8, 10 and 14, but B12 = −691/2730 gives c12 = 65520/691, so E12 = 1 + (65520/691) Σ σ11(n) qn has 691 in the denominator of its q-coefficient and of every later one up to n = 1380. (The exceptions are the n with 691 dividing σ11(n). The first is the prime n = 1381, which is −1 mod 691, and any n divisible exactly once by such a prime is another. They are rare early on, about 1 in 700 up to n = 200 000, but asymptotically they make up almost all n.) That prime returns in Part 7 as Ramanujan's congruence. Over ℤ, E4 and E6 are not enough: Part 5 shows that the forms with integer q-expansions need Δ as a third generator, with the single relation E43 − E62 = 1728Δ.
The lattice sum makes modularity clear but not the arithmetic. The q-expansion shows the arithmetic (divisor sums, integrality, multiplicativity) but not the modularity. Lipschitz summation connects them, and the same split comes back 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.