**5**

votes

**1**answer

247 views

### Sums of random variables mod p

Let $\varepsilon_1, \ldots, \varepsilon_n$ be independent random variables taking values $0,1$ each with probability $1/2$. It is well known that $R_n=\varepsilon_1+ \cdots+ \varepsilon_n$ modulo a ...

**3**

votes

**0**answers

231 views

### Best known bounds on certain exponential sums

What are the best bounds currently known for the following exponential sum:
$$\sum_{x < p \le 2x} e(\alpha p^k)$$
for values of $\alpha$ far from a rational with small denominator. ($p$ refers ...

**9**

votes

**2**answers

539 views

### Iwaniec-Kowalski Exponential Sum for Quadratic Function

I am reading about 'Exponential Sums' in the book 'Analytic Number Theory' by Iwaniec and Kowalski. On page 199 they mention the bound:
$$|S_f(N)|^2 \le N +2N^2q^{-1}+4(N+q)\log q \tag{1}$$
where, ...

**9**

votes

**1**answer

275 views

### $L^1$ norm of exponential sum of $n^2 x$

What is the asymptotic order of
$$
\int_0^1 \left| \sum_{n=1}^N e^{2 \pi i n^2 x} \right| ~dx
$$
as $N \to \infty$. This should be known, but I cannot find it in the literature.

**0**

votes

**1**answer

68 views

### What is the maximal number of solutions of $\sum_{i = 1}^n 1/a_i^x - \sum_{i = 1}^m 1/b_i^x = 0$?

What is the maximal number of solutions of the following equation?
$\sum_{i = 1}^n 1/a_i^x - \sum_{i = 1}^m 1/b_i^x = 0$
where $x$ is the unknown and $n, m$, $a_i$'s, $b_i$'s are constant.
It ...

**1**

vote

**1**answer

108 views

### Uniform convergence of infinite sum with Dirichlet characters

I would like to prove uniform convergence of function series like :
$$\sum\limits_{n=1}^{\infty} \chi(n) f(nx)$$ where $\chi$ is a primitive character and $f(x)$ a function decreasing to zero in ...

**1**

vote

**3**answers

141 views

### Decidability of sum of powers exponential diophantine equation

I want to ask about decidability of exponential Diophantine equation:
$z_12^{\eta_1} + \ldots + z_n2^{\eta_n} = z$, where $z_i,\eta_i,z \in \mathbb{Z}$, and $\eta_i$ - are variables.
Can we find ...

**0**

votes

**0**answers

67 views

### separating parameters in generalized quadratic Gauss sum

The normalized generalized quadratic Gauss sum is defined by
$$
G(a,b,c)=\frac{1}{c}\sum_{n=1}^ce\left(\frac{an^2+bn}{c}\right)
$$
where $e(x)=\exp(2\pi ix)$.
Under what conditions on $c$ can we ...

**3**

votes

**0**answers

226 views

### Solving a doubly exponential generating function

I am analyzing the average time complexity of some algorithm on some probabilistic model, and I've come to a doubly exponential sequence for which I cannot find corresponding generating function. I ...

**2**

votes

**1**answer

135 views

### Bounds on imaginary parts of partial Kloosterman sums?

For a prime $p$ and integers $a,m$, $0<a,m<p$ define the (partial Kloosterman) sum
$$ S_p(a,m) = \sum_{0<k<m} \exp\left(\frac{2\pi\mathrm{i}}{p}(a x + x^{-1})\right), $$
where $x^{-1}$ is ...

**2**

votes

**2**answers

212 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 ...

**0**

votes

**0**answers

52 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

**2**answers

510 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

**0**answers

162 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

**1**answer

455 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

576 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

630 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 ...

**14**

votes

**1**answer

780 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

**1**answer

447 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

262 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 ...