# Questions tagged [gauss-sums]

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

83 questions
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### Sub-Sum of Roots of Unity

Let $\alpha$ be an algebraic integer of a cyclotomic field, and let $\theta_1, \theta_2, ..., \theta_n$ be roots of unity such that $$\sum_{i=1}^n \theta_i = 2\alpha.$$ Does there necessarily exists a ...
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### Can a bound be given to $\sum_{a=1}^{p-1}\chi(a)\left(\frac{a^2-1}{p}\right)$, which is smaller than $p$, where $\chi$ is a dirichlet character?

Here $(\cdot)$ denotes the legendre symbol and $p$ be an odd prime number. All terms are of norm 1. So one bound is $p$. can one better bound be given to the sum?
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### Expected maximum of sub-Gaussian

I'm trying to answer the following question from the book high-dimensional probability: Let $X_1,X_2,\dots$ be a sequence of sub-gaussian random variables, which are not necessarily independent. Show ...
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### How to prove that below quantity is purely imaginary?

How do I prove that the following quantity is purely imaginary: $$\sum_{0\leq l_1<l_2<l_3<l_4\leq q-1} e^{-2\pi i \frac{(l_1^2-l_2^2+l_3^2-l_4^2)}{q} }$$ where $q$ is an odd number?
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### Finding closed form of a summation $\sum_{i = \lfloor{\frac{n}{2}}\rfloor}^{n}i$

So i'm trying to find the clsoed form of $\sum_{i = \lfloor{\frac{n}{2}}\rfloor}^{n}i$ from what I know its probably going to be $? * (\lfloor{\frac{n}{2}}\rfloor + n)$ based on when I wrote out ...
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### Infinite quadratic gauss sum.

I have a summation of the form: $$\sum_{m\in\mathbb{Z}}m\exp\left\{-\frac{B}{2}\left(z-\frac{2\pi iv}{B}-\frac{2\pi m}{BL_2}\right)^2\right\}$$ where $B, v, L_2$ are constants and $z$ is a variable. ...
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### Gauss sum variation $\sum_{n=0}^{p-1}\left(\frac{a+bn}{p}\right)\zeta_p^{cn} = ?$

I'm having trouble evaluating this, for $a, b, p$ all pairwise coprime, $p$ an odd prime, $c$ any integer. $$\sum_{n=0}^{p-1}\left(\frac{a+bn}{p}\right)\zeta_p^{cn}$$ Any help/references would be ...
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### Upper bound on the summation of roots of unity

Let $p$ be a prime number, and $\zeta = e^{(2\pi i/p)}$ be a $p$th root of unity. Given the gauss sum: $$g_t = \sum_{k = 0}^{p-1}\left(\frac{k}{p}\right)\zeta^{tk}$$ I'm trying to prove the upper ...
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### How to prove a generalized Gauss sum formula

I read the wikipedia article on quadratic Gauss sum. link First let me write a definition of a generalized Gauss sum. Let $G(a, c)= \sum_{n=0}^{c-1}\exp (\frac{an^2}{c})$, where $a$ and $c$ are ...
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### Gauss formula to add number of sequence for arbitrary range

Gauss formula to add numbers from $1-100$ is: $$\frac{n(n+1)}{2}$$ How can this be made applicable for arbitrary range, lets say $3-30$? Is there an easy way of doing that rather then linearly ...
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### How is distributed the sum of square of correlated z-score?

I have a $1\times k$ matrix representing $z$-scores and each element is correlated to each other according to a covariance matrix $\Sigma$. I would like to compute their sum of square and to know the ...
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### Product of Gauss sums $\tau_p,\tau_q$

Let $p,q$ be different odd primes, and let $\tau_p = \sum\limits_{a=1}^{p}\left(\frac{a}{p}\right)e^{\frac{2\pi ia}{p}}$, $\tau_p = \sum\limits_{b=1}^{q}\left(\frac{b}{q}\right)e^{\frac{2\pi ib}{q}}$. ...
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### Upper bound for the multidimensional gaussian integral

How to prove that, for any integer $d \geq 1$ and any $\alpha > 1/2$, $$\int_{n^{\alpha}}^{\infty} r^{d-1} e^{-\frac{r^2 d}{n}} dr$$ goes to $0$ with $n$? This can be interpreted (up to a ...
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### Cross-correlation of Gaussian and Jacobian sums

I recently came upon the following kind of sum and I'm wondering if anyone has seen it before, or could point out something interesting about them. Let $F$ be a finite field with $q > 2$ elements ...
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