Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 137 / 4 / b

Table of contents
- Solution b

Solution

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}) .

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version