# Questions tagged [gauss-sums]

For questions on Gauss sums, a particular kind of finite sum of roots of unity.

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### Gaussian like sum with binomial coefficients

Is there any way to calculate expressions of the form $$f(q)=\sum_{n=0}^N\binom{n}{k}\exp({2\pi i\tfrac{n}{N}\cdot q})$$ where $q\in\mathbb{Q}$? It reminds me quite alot of Gauss sums, but I don't ...
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### Understanding the proofs in "Elementary aspects of the Verlinde formula and of the Harder–Narasimhan–Atiyah–Bott formula" from Don Zagier

As the proofs of Theorem 1 in "elementary aspects of the Verlinde formula and of the Harder-Narasimhan-Atiyah-Bott formula" are very terse I could not understand all proofs. The paper can be ...
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### Sum of even numbers N? [closed]

To sum first N numbers we can use this formula: $$1 + 2 + 3 + \ldots + n = \frac{(n (n + 1)}{2}$$ To sum even numbers we multiply this formula by 2: $$2 + 4 + 6 + \ldots + 2n = n (n + 1)$$ Lets ...
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Let $\chi$ be a Dirichlet character modulo $q$. Let $e(x)=\text{exp}(2\pi i x)$ and $\tau(\chi)$ be the Gauss sum $\tau(\chi)=\sum_{m \,\text{mod}\, q}\chi(m)e(m/q)$. For any $n$ with $\gcd(n,q)=1$, ...