Math 2.0

Number theory Definition

Dedekind eta function

Builds on Upper half-planeHolomorphic functionExponential function

Definition

Let ℍ\mathbb{H} be the upper half-plane, and for τ∈ℍ\tau \in \mathbb{H} let q=e2πiτq = e^{2 \pi i \tau}, so that ∣q∣<1\abs{q} < 1. The Dedekind eta function is the function η ⁣:ℍ→ℂ\eta \colon \mathbb{H} \to \C given by η(τ)=eπiτ/12∏n=1∞(1−qn).\eta(\tau) = e^{\pi i \tau / 12} \prod_{n = 1}^{\infty} (1 - q^n).

Remarks

The product converges absolutely and locally uniformly, since ∑n∣q∣n<∞\sum_n \abs{q}^n < \infty. So η\eta is holomorphic on ℍ\mathbb{H}, and it has no zeros there. It satisfies η(τ+1)=eπi/12η(τ)\eta(\tau + 1) = e^{\pi i / 12} \eta(\tau) and η(−1/τ)=−iτ η(τ)\eta(-1/\tau) = \sqrt{-i\tau}\, \eta(\tau), with the principal branch of the square root. Example: η(τ)24=q∏n≥1(1−qn)24=Δ(τ)\eta(\tau)^{24} = q \prod_{n \ge 1} (1 - q^n)^{24} = \Delta(\tau), the cusp form of weight 1212. Non-example: η\eta itself is not invariant under τ↦τ+1\tau \mapsto \tau + 1, because of the factor eπi/12e^{\pi i / 12}, so it is not a modular form for SL2(ℤ)\mathrm{SL}_2(\Z). By the pentagonal number theorem, ∏n≥1(1−qn)=∑m∈ℤ(−1)mqm(3m−1)/2\prod_{n \ge 1} (1 - q^n) = \sum_{m \in \Z} (-1)^m q^{m(3m - 1)/2}.

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  1. Complex numbers object
  2. Exponential function concept
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  4. Holomorphic function concept
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  7. Möbius transformation concept
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  9. Square matrix definition
  10. Linear map definition
  11. Multilinear map definition
  12. Alternating multilinear map definition
  13. Determinant definition
  14. Special linear group object
  15. Upper half-plane object
  16. Dedekind eta function definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Dedekind_eta_function
About this page
TagKVZ0VZ6W01
URIm2://KVZ0VZ6W01
Kinddefinition
WritingW0 stub
MSC11F20
DefinesDedekind eta function
UsesUpper half-plane, Holomorphic function, Exponential function
Changed2 November 2026

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