Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $K$ be a field (not necessarily $\mathbb C$) and let $\zeta=\zeta_n$ be a primitive $n$th root of unity in $\bar K$.

I would like to know if there is a formula calculating $$ \sum_{k=1}^n \zeta^{-k^2}.$$

If it exists please tell me the formula.

Thank you in advance.

share|cite|improve this question
Experiments suggest that if $n\equiv 1 \pmod{4}$ then the sum is $\pm\sqrt{n}$ and if $n\equiv 3 \pmod{4}$ then the sum is $\pm\sqrt{n}i.$ – minar Aug 27 '13 at 20:33
Related question:… – minar Aug 27 '13 at 20:43
I would like to consider this for a general filed $K$ not necessarily $mathbb C$. Is the same computation true? – Snow Aug 27 '13 at 21:47
Since it is in fact true over $\bar{\mathbb{Q}}$, the computation should transfert to quite a lot of cases, including finite fields or $p$-adic. – minar Aug 28 '13 at 7:36

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.