The method of exponential sums is one of a few general methods enabling us to solve a wide range of miscellaneous problems from the theory of numbers and its applications. The strongest results have been obtained with the aid of this method. Therefore knowledge of the fundamentals of theory of ...

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2
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2answers
170 views

Bound on exponential sum with weights

Let $e(z)$ denote $e^{2 \pi i z}$ and let $f(z)$ a smooth real function. I know one can bound sums of the form $$ \sum_{x \leq X} e(f(x)) $$ via for example Van der Corputs's result, provided we make ...
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0answers
47 views

Definition of Degenerate hyper-Kloosterman Sum

If $(a,q)=1$, we all know the usual hyper-Kloosterman sum $$Kl_n(a,q)= \sum_{x_1...x_n\equiv a \mod q} e(\frac{x_1+\dots+x_n}{q}).$$ I wonder what would be the correct definition when $(a,q)\neq 1$. ...
8
votes
2answers
438 views

Incomplete Kloosterman sum

I am interested in an upper bound on the following incomplete Kloosterman sum $$ \sum_{\substack{x=1 \\ x+_{_{\bf Z}}x^{-1}>p}}^{p-1}e\left(\frac{x+x^{-1}}{p}\right).$$ Using the Weil's bound it ...
2
votes
0answers
153 views

Kloosterman-like sum with inverse to different moduli

In some recent work, the following strange-looking exponential sum arose: $$ \sum_k \sum_r \sum_s e\bigg( \frac{r \bar s^{(r)} \bar k^{(r^2+s^2)}}{r^2+s^2} \bigg). $$ Here $e(x) = e^{2\pi i x}$ as ...
1
vote
1answer
409 views

stationary phase method in analytic number theory

I hope someone can tell me something about the error term in the formula calculating the oscillatory integral like $\int_a^b g(x)e(f(x))d x$. Specially, the exact formula on page 114 of M. Huxley's ...
1
vote
1answer
548 views

Exponential sums

I would like to estimate the following sum $\sum_{N <n \leq 2N}e(vn^{l})$, $l \geq 1$ constant(not integer) and $v$ is a parameter(integer) that doesn't grow too fast(a small power of N). The ...
21
votes
1answer
1k views

Is there a cheap proof of power savings for exponential sums over finite fields?

Let $p$ be a large prime, and let $f(x) = P(x)/Q(x)$ be a non-constant rational function over ${\Bbb F}_p$ of bounded degree. From the Weil conjectures for curves, we have a bound of the form $$ ...
3
votes
3answers
539 views

iteratively (approximately) solving a sum of exponentials

I would iteratively have to solve the following equation at iteration $n$: $C = \sum_{1 \leq i \leq n}{e^{\frac{x_i}{T}}x_i}$ for $T$. Each iteration $i$ an unknown $x_i$ will be observed and $C$ is ...
13
votes
1answer
663 views

On the $L^1$-norm of certain exponential sums.

I am stuck with an elementary-looking problem, which does not belong to my usual field of research so I eventually decided to ask it on MO. Let $S$ be a finite set of integers. For $P$ a subset of ...
7
votes
1answer
385 views

Lower bound for exponential sums.

Let $D$ be a subset of $\mathbb Z/n \mathbb Z$ containing $0$. For $m$ an integer, set $$\alpha(m,D)=\sum_{d \in D} e\left (\frac{m d }{n}\right ),$$ where as usual $e(x) = e^{2 i \pi x}$ This is an ...
5
votes
1answer
246 views

Where can I read about exponential sums corresponding to Jones Polynomial?

I remember reading that a number theoretic analogue of Witten's path integral formula for the Jones polynomial: $$\text{Jones}_K(e^{2\pi i/(k+2)})=\int_{\text{$SU(2)$ connections on $\mathbb ...