**1**

vote

**1**answer

331 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

**1**answer

513 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

**1**answer

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

**3**answers

453 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 ...

**12**

votes

**1**answer

586 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 ...

**6**

votes

**1**answer

321 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

**1**answer

234 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 ...