All Questions
Tagged with exponential-sums nt.number-theory
134 questions
2
votes
1
answer
304
views
Exponential sum (linear in the argument) over primes
Suppose we have $\alpha \in \mathbb{R}$. Then we know that
$$\sum_{1 \leq n \leq X} e(n \alpha) \ll \min \{ X, \|\alpha\|^{-1} \}$$
where $\| \cdot \|$ is the distance to the nearest integer.
I ...
- 3,625
4
votes
0
answers
158
views
Unique factorization for the semigroup generated by {2cos(π/n) | n>3}?
Let $S$ be the multiplicative semigroup of numbers generated by $B=\{ 2cos(\frac{\pi}{n}) \mid n \ge 4 \}$.
Question: Does every number of $S$ factorize uniquely (up to perm.) as a product of ...
4
votes
1
answer
307
views
When the Kloosterman sum is an integer?
Let $q$ be a power of prime $p$ and $\zeta_p$ be the complex $p$ th root
of unity. We denote by
$\mathbb{F}_q$ the finite field of $q$ elements and by $Tr$ the absolute trace function $\mathbb{F}_q\...
- 255
1
vote
1
answer
346
views
On a sum like Kloosterman sum
I encounter a tricky sum like the Kloosterman sum
$$\sum_{x \mod qP} e ( \frac{x+\overline{x+P}}{qP} ),$$
where $q$ is a positive integer, $P$ is a prime number satisfying $(q,P)=1$, $x \bmod qP,(x,...
- 353
1
vote
0
answers
165
views
Reference request: Bounding exponential sum $\sum_{x \in [0,X]} \Lambda(x) e(\beta_d x^d + \ldots + \beta_1 x )$
Let $1 \leq i \leq d$, $q \in \mathbb{N}$, and $0 \leq a_{i} < q$. Let
$$
\mathfrak{M}^{(i)}_{a_{ i}, q} (C) =\{ \beta_{i} \in [0,1) : | \beta_{i} - a_{i}/q | \leq (\log X)^{C} X^{-i} \} .
$$
We ...
- 3,625
3
votes
0
answers
263
views
Number of solutions to $x_1x_2=x_3x_4\bmod n$
In https://www.math.ksu.edu/~cochrane/research/xyuvmodp.pdf it is shown $x_1x_2=x_3x_4\bmod p$ where $p$ is a prime has $\frac{|\mathcal B|}p+O(\sqrt{|\mathcal B|}\log^2p)$ solutions $(x_1,x_2,x_3,...
- 13.9k
5
votes
1
answer
392
views
Does anyone recognize this exponential sum?
For $a$, $b$ two integers, let $(a,b)$ denotes their gcd. We define the following exponential sum :
$$G_q(n):=\sum_{d|q,~(d,q/d)=1}{e^{2i\pi n\frac{dd'}{q}}}$$
for $n$ a non-negative integer and $q$ ...
- 1,479
2
votes
0
answers
330
views
On exponential sum weighted with von-Mangoldt function
Suppose we have $\alpha \in \mathbb{R}$ such that $|\alpha - a/q| < 1/q^2$,
where $(a,q)=1$. Then we know that the exponential sum
$$
S(\alpha) = \sum_{1 \leq n \leq X} \Lambda(n) e(n \alpha)
$$
...
- 3,625
4
votes
1
answer
776
views
An exponential sum over squares
I have the following exponential sum:
$\sum _{M<n\leq N}e\left (x/n^2\right )=\sum f(n),$
say, where $M$ and $N$ are something like $x^{1/4}$ and $x^{1/2}$.
My question is basically, how do I ...
- 1,381
16
votes
1
answer
1k
views
On (a generalization of) the Gauss Circle Problem
Most (if not all) references I read about the Gauss Circle Problem that proves a bound below $O(R^{2/3})$ reduces the GCP to the Dirichlet Divisor Problem by the well known expression of $r_2(n)$, the ...
- 5,169
1
vote
1
answer
255
views
Exponential sum estimates similar to the one for $\sum_p (\log p) e(p \alpha)$, but for different sequences
Obtaining a non-trivial estimate for $\sum_p (\log p) e(p \alpha)$ over the minor arcs is one of the estimates required for obtaining the ternary Goldbach for $n$ sufficiently large via the circle ...
- 579
9
votes
1
answer
729
views
Sums of twisted products of Kloosterman Sums
For $m,n,c \in \mathbb{N}$, let $S(m,n;c)$ denote the Kloosterman sum
$$
S(m,n;c) := \sum_{\substack{1 \leq a < c \\ \gcd(a,c) = 1}} e \left( \frac{ma + n\overline{a}}{c} \right)
$$
where $e(n) = e^...
- 1,570
2
votes
1
answer
304
views
Kloosterman sum
Does anybody know the non-trivial bound for this sum?
$S(m,n,c,q)=\sum_{a,b\in \mathbb{Z}/cq\mathbb{Z}, \;ad\equiv 1\text{ mod }c} e^{2\pi i(am+nd)/qc},$
where $m,n\in\mathbb{Z},\;q,c\in\mathbb{N}$.
...
- 21
10
votes
2
answers
1k
views
Bounding exponential sum with square roots
It is well known that for each $m\in\mathbb{N}$
$$\lim_{N\to\infty}\frac1N\sum_{n=1}^Ne^{2\pi i\sqrt{nm}}=0$$
My question is whether there is some uniformity in the variable $m$.
More precisely, is it ...
- 1,701
6
votes
1
answer
285
views
What is the mean value of a pair of Ramanujan Sums when summed over squares?
Does anyone know of the mean value of two Ramanujan Sums when summed over the square of integers?
In my research on the Landau problem regarding nearly square primes, I have run into the mean value ...
- 221
17
votes
1
answer
593
views
Smoothed exponential sums: bounds and sources?
Let $f:\mathbb{R}\to\mathbb{C}$ be differentiable $k$ times, with $f, f',\dotsc,f^{(k)}\in L^1$. Let $\alpha\in \mathbb{R}/\mathbb{Z}$, $\alpha\ne 0$. In "Every odd number..." (Math. Comp. 83, 2014), ...
- 20.2k
8
votes
2
answers
675
views
The number of solution of $x_1^2 + \cdots + x_k^2 \equiv \lambda \bmod q$
I'm playing with exponential sums...
If $q$ is an odd prime and $a$ an integer such that $q \nmid a$, then the following formula for the Gaussian sum is known
$$\sum_{x=0}^{q-1} e_q(ax^2) = \left(\...
user40023
3
votes
1
answer
382
views
Twisting by a multiplicative Character in Katz, Perversity and Exponential sums
Let $C(x_1,\ldots,x_n)$ be a nonsigular cubic form with integral coefficients.
In his proof that $C$ fulfills the Hasse-Principle, if $n\geq 9$, Hooley used the following estimate that was provided by ...
- 33
4
votes
0
answers
562
views
Best known bounds on certain exponential sums
What are the best bounds currently known for the following exponential sum:
$$\sum_{x < p \le 2x} e(\alpha p^k)$$
for values of $\alpha$ far from a rational with small denominator. ($p$ refers ...
- 1,890
11
votes
2
answers
2k
views
Iwaniec-Kowalski Exponential Sum for Quadratic Function
I am reading about 'Exponential Sums' in the book 'Analytic Number Theory' by Iwaniec and Kowalski. On page 199 they mention the bound:
$$|S_f(N)|^2 \le N +2N^2q^{-1}+4(N+q)\log q \tag{1}$$
where, $...
- 810
9
votes
1
answer
564
views
$L^1$ norm of exponential sum of $n^2 x$
What is the asymptotic order of
$$
\int_0^1 \left| \sum_{n=1}^N e^{2 \pi i n^2 x} \right| ~dx
$$
as $N \to \infty$. This should be known, but I cannot find it in the literature.
- 2,207
1
vote
1
answer
262
views
Uniform convergence of infinite sum with Dirichlet characters
I would like to prove uniform convergence of function series like :
$$\sum\limits_{n=1}^{\infty} \chi(n) f(nx)$$ where $\chi$ is a primitive character and $f(x)$ a function decreasing to zero in ...
- 1,199
1
vote
3
answers
284
views
Decidability of sum of powers exponential diophantine equation
I want to ask about decidability of exponential Diophantine equation:
$z_12^{\eta_1} + \ldots + z_n2^{\eta_n} = z$, where $z_i,\eta_i,z \in \mathbb{Z}$, and $\eta_i$ - are variables.
Can we find ...
- 11
3
votes
1
answer
229
views
Bounds on imaginary parts of partial Kloosterman sums?
For a prime $p$ and integers $a,m$, $0<a,m<p$ define the (partial Kloosterman) sum
$$ S_p(a,m) = \sum_{0<k<m} \exp\left(\frac{2\pi\mathrm{i}}{p}(a x + x^{-1})\right), $$
where $x^{-1}$ is ...
- 416
2
votes
2
answers
763
views
Bound on exponential sum with weights
Let $e(z)$ denote $e^{2 \pi i z}$ and let $f(z)$ a smooth real function.
I know one can bound sums of the form
$$
\sum_{x \leq X} e(f(x))
$$
via for example Van der Corputs's result, provided we make ...
- 579
11
votes
2
answers
1k
views
Incomplete Kloosterman sum
I am interested in an upper bound on the following incomplete Kloosterman sum
$$ \sum_{\substack{x=1 \\ x+_{_{\bf Z}}x^{-1}>p}}^{p-1}e\left(\frac{x+x^{-1}}{p}\right).$$
Using the Weil's bound it ...
- 499
2
votes
0
answers
227
views
Kloosterman-like sum with inverse to different moduli
In some recent work, the following strange-looking exponential sum arose:
$$
\sum_k \sum_r \sum_s e\bigg( \frac{r \bar s^{(r)} \bar k^{(r^2+s^2)}}{r^2+s^2} \bigg).
$$
Here $e(x) = e^{2\pi i x}$ as ...
- 12.8k
1
vote
1
answer
745
views
stationary phase method in analytic number theory
I hope someone can tell me something about the error term in the formula calculating the oscillatory integral like $\int_a^b g(x)e(f(x))d x$. Specially, the exact formula on page 114 of M. Huxley's ...
- 177
1
vote
1
answer
767
views
Exponential sums
I would like to estimate the following sum
$\sum_{N <n \leq 2N}e(vn^{l})$, $l \geq 1$ constant(not integer) and $v$ is a parameter(integer) that doesn't grow too fast(a small power of N).
The ...
- 167
32
votes
1
answer
3k
views
Is there a cheap proof of power savings for exponential sums over finite fields?
Let $p$ be a large prime, and let $f(x) = P(x)/Q(x)$ be a non-constant rational function over ${\Bbb F}_p$ of bounded degree. From the Weil conjectures for curves, we have a bound of the form
$$ |\...
- 114k
12
votes
2
answers
1k
views
counting points on unit sphere mod p
Let $f(n)$ be the number of points on the unit sphere $x^2 + y^2 + z^2 = 1\; \pmod n$ with $x,y,z \in \mathbb{Z}/n\mathbb{Z}$
This is sequence A087784 in the Online Encyclopedia of Integer ...
- 22.8k
14
votes
1
answer
2k
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 $S$,...
- 26k
11
votes
1
answer
1k
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 ...
- 26k
5
votes
1
answer
374
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 $\mathbb S^3$}/\...
- 18.7k