Questions tagged [character-sums]

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2 votes
1 answer
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Finding a certain value of $\Gamma_p$

Let $\Gamma_p : \mathbb{Z}_p \to \mathbb{Z}_p^{\times}$ be the $p$-adic gamma function. I thought that I had successfully calculated $\Gamma_p(1 - 1/4)$, but sage is telling me I'm wrong (this is ...
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2 votes
2 answers
152 views

Value of the quadratic Gauss sum viewed in $\mathbb{C}_p$

Let $p > 2$ be a prime and $q = p^r$ for some $r \in \mathbb{Z}^+$. I will assume that all roots of unity lie in $\mathbb{C}_p^{\times}$. Let $\zeta$ a primitive $p$-th root of unity. Let $Tr : ...
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6 votes
0 answers
89 views

Bounds on exponential and character sums of ratio of linear recurrences

Let $\mathbb{F}_q$ be a finite field of $q$ elements, let $\chi$ be a non-trivial additive character of $\mathbb{F}_q$, and let $\psi$ be a non-trivial multiplicative character of $\mathbb{F}_q$. Also,...
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2 votes
0 answers
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Expected order of magnitude of character sums under GRH

Let $\chi$ be a nonprincipal character with modulus $q$. Under GRH, what is the expected order of magnitude of $\sum_{n \le x} \chi(n)$, where I think of $x$ and $q$ as growing, but $x$ is smaller ...
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8 votes
0 answers
349 views

Prime character sums

Let $p$ be a (large) prime number, and let $\chi : (\mathbf{Z}/p\mathbf{Z})^{\times} \rightarrow \mathbf{C}^{\times}$ be a Dirichlet character of conductor $p$. We have good estimates on the character ...
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3 votes
1 answer
604 views

Sum of Legendre symbol over primes

Given some $X, Y\ge 1$ and some $d\le Y$ not a perfect square, is it possible to bound $$\sum_{p\le X}\left(\frac{d}{p}\right)?$$ As long as $Y$ is not too large compared to $X$, I would expect that ...
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5 votes
0 answers
137 views

Character sums over a sumset

Suppose that $p$ is a prime, $A$ is a subset of $\mathbb F_p$, and $P$ is a polynomial over $\mathbb F_p$ of degree $d$. Using Weil's bound, it is not difficult to show that $$ \left| \sum_{a,b\in A}...
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4 votes
1 answer
134 views

About Averages of Incomplete Additive Character Sums

Let $p$ be a prime. Let $\omega$ be a $ p $-th root of unity. We know that $ \chi_\alpha (x) = \omega^{\alpha\cdot x} $ are the additive characters of $ \mathbb{Z}_p $. I have a question about ...
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