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 Incoming links: Weighted Gaussian theta sum of a complex lattice

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 137 / 4 / b / Solution Created 2026-09-28 Updated 2026-10-03

Apply part a in

R^{2}

to

f_{k} (x, y) = (x + i y)^{k} e^{- π (x^{2} + y^{2})} .

The supplied identity

f_{k} = (- i)^{k} f_{k}

gives

\sum_{λ \in Λ} λ^{k} e^{- π ∣ λ ∣^{2}} = (- i)^{k} m (Λ)^{- 1} \sum_{μ \in Λ^{\lor}} μ^{k} e^{- π ∣ μ ∣^{2}} .

In the notation of the weighted Gaussian theta sum of a complex lattice, this is

θ_{k} (Λ) = (- i)^{k} m (Λ)^{- 1} θ_{k} (Λ^{\lor}) .

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