Skip to main content
MathOverflow

Questions tagged [character-sums]

Ask Question

The tag has no summary.

13 questions
Newest Active Bountied Unanswered
2 votes
1 answer
123 views

Let $k$ be fixed. Let $q_1,q_2,\dotsc,q_k$ be coprime with product $\prod_{j=1}^k q_j$ of size about $N$. Let $n$ range over integers in $[1,\sqrt{N}]$ coprime to $q_1,q_2,\dotsc,q_k$. Is the vector $$...
H A Helfgott's user avatar
  • 22k
2 votes
0 answers
202 views

Let $q,r\in \mathbb{N}$, and $\chi$ be primitive Dirichlet character mod $q$ with $(q,r)\neq 1$, $(b,r)=1$. $F$ is a smooth function. Then what we will get after applying the Poisson summation on the ...
RGM's user avatar
  • 49
1 vote
0 answers
256 views

Let $\chi$ be a quadratic character mod $q$. I am interested in finding the best result for how large $N$ should be such that it is guaranteed that $$\sum_{p=1}^{N} \chi(p) \log p= o(N).$$ I am aware ...
Farzad Aryan's user avatar
  • 499
29 votes
1 answer
3k views

Let $n>1$ be a natural number such that $2n-1$ is not divisible by $3$, let $\chi: {\bf Z} \to {\bf C}$ be a non-principal Dirichlet character of period $2n-1$, and let $\omega := e^{2\pi i/3}$. ...
Terry Tao's user avatar
  • 120k
0 votes
1 answer
208 views

This is a question on the paper: D. R. Heath-Brown. Artin's conjecture for primitive roots. Quarterly J. Math. 37 (1): 27–38, 1986. At the beginning of the proof of his main theorem on page 35, Heath-...
David R's user avatar
  • 3
2 votes
1 answer
261 views

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 ...
matt stokes's user avatar
  • 727
2 votes
2 answers
388 views

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 : ...
matt stokes's user avatar
  • 727
6 votes
0 answers
132 views

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,...
Erik4's user avatar
  • 121
3 votes
0 answers
132 views

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 ...
Ofir Gorodetsky's user avatar
  • 16.5k
11 votes
0 answers
662 views

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 ...
Emmanuel Lecouturier's user avatar
  • 1,139
3 votes
1 answer
1k views

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 ...
Mayank Pandey's user avatar
  • 1,909
5 votes
0 answers
174 views

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}...
Seva's user avatar
  • 23.5k
4 votes
1 answer
181 views

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 ...
Akshay's user avatar
  • 43