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Stream: learning: questions

Topic: Urs Schreiber's question about a trig identity

Madeleine Birchfield (Mar 15 2025 at 14:47):

Urs Schreiber asked this question on the nForum:

This may just show my ignorance, but I am wondering how to see the following:

For $k \in \mathbb{N}_{> 0}$ and $a \in \mathbb{N}$, the expression

$$S_k(a) := \sum_{n=0}^{k−1} e^{\frac{πi}{k} (n+a)^2}$$

is independent of a if k is even:

$$k \; \mathrm{is} \; \mathrm{even} \; \Rightarrow \forall_a(S k(a)=S k(0)).$$

In a roundabout way this follows from the modular transformation law of conformal characters of the U(1) WZW model.

But what’s an elementary arithmetic way to see this?

(If the exponent had another factor of “2”, then it would follow easily, also for odd $k$, by modular arithmetic, now in the other sense of “modular”. But the question is for the exponent as given.)

https://nforum.ncatlab.org/discussion/19005/a-trig-identity/