All Questions
Tagged with exponential-sums nt.number-theory
134 questions
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
$$ |\...
- 114k
17
votes
1
answer
593
views
Smoothed exponential sums: bounds and sources?
Let $f:\mathbb{R}\to\mathbb{C}$ be differentiable $k$ times, with $f, f',\dotsc,f^{(k)}\in L^1$. Let $\alpha\in \mathbb{R}/\mathbb{Z}$, $\alpha\ne 0$. In "Every odd number..." (Math. Comp. 83, 2014), ...
- 20.2k
16
votes
1
answer
1k
views
On (a generalization of) the Gauss Circle Problem
Most (if not all) references I read about the Gauss Circle Problem that proves a bound below $O(R^{2/3})$ reduces the GCP to the Dirichlet Divisor Problem by the well known expression of $r_2(n)$, the ...
- 5,169
14
votes
1
answer
2k
views
On the $L^1$-norm of certain exponential sums
I am stuck with an elementary-looking problem, which does not belong to my usual field of research so I eventually decided to ask it on MO.
Let $S$ be a finite set of integers. For $P$ a subset of $S$,...
- 26k
13
votes
2
answers
800
views
For which rationals is this exponential sum bounded?
Given $x \in [0, 1]$, we denote by $e(x)$ the complex number $e^{2 \pi i x}$.
Can we characterise the set of rationals $x$ for which the sum
$$A_N(x)\, :=\, \sum_{n = 0}^N e(2^n x)$$
remains bounded ...
- 6,155
12
votes
2
answers
1k
views
counting points on unit sphere mod p
Let $f(n)$ be the number of points on the unit sphere $x^2 + y^2 + z^2 = 1\; \pmod n$ with $x,y,z \in \mathbb{Z}/n\mathbb{Z}$
This is sequence A087784 in the Online Encyclopedia of Integer ...
- 22.8k
11
votes
2
answers
2k
views
Iwaniec-Kowalski Exponential Sum for Quadratic Function
I am reading about 'Exponential Sums' in the book 'Analytic Number Theory' by Iwaniec and Kowalski. On page 199 they mention the bound:
$$|S_f(N)|^2 \le N +2N^2q^{-1}+4(N+q)\log q \tag{1}$$
where, $...
- 810
11
votes
2
answers
1k
views
Incomplete Kloosterman sum
I am interested in an upper bound on the following incomplete Kloosterman sum
$$ \sum_{\substack{x=1 \\ x+_{_{\bf Z}}x^{-1}>p}}^{p-1}e\left(\frac{x+x^{-1}}{p}\right).$$
Using the Weil's bound it ...
- 499
11
votes
1
answer
1k
views
Lower bound for exponential sums
Let $D$ be a subset of $\mathbb Z/n \mathbb Z$ containing $0$. For $m$ an integer, set $$\alpha(m,D)=\sum_{d \in D} e\left (\frac{m d }{n}\right ),$$
where as usual $e(x) = e^{2 i \pi x}$ This is an ...
- 26k
11
votes
1
answer
866
views
Cancellation in a very rapidly oscillating exponential sum
Let $f(x)$ be a function such that for all $T \leq f(x), \ T / x \to \infty$ we have
$$
\sum_{x \leq n < 2 x} e^{2 \pi i T / n} = o(x).
$$
How fast can $f(x)$ grow?
I can show that for any $\...
- 2,404
10
votes
2
answers
1k
views
Bounding exponential sum with square roots
It is well known that for each $m\in\mathbb{N}$
$$\lim_{N\to\infty}\frac1N\sum_{n=1}^Ne^{2\pi i\sqrt{nm}}=0$$
My question is whether there is some uniformity in the variable $m$.
More precisely, is it ...
- 1,701
10
votes
1
answer
474
views
A basic estimate of exponential sums
Demeter in his book "Fourier Restriction, Decoupling, and Applications" (P287) used the following estimate:
\begin{equation}
\sup_{0\leq n\leq q}\bigg|\sum_{m=0}^n e^{2\pi i\frac{a}{q}m^2}\...
- 351
9
votes
1
answer
564
views
$L^1$ norm of exponential sum of $n^2 x$
What is the asymptotic order of
$$
\int_0^1 \left| \sum_{n=1}^N e^{2 \pi i n^2 x} \right| ~dx
$$
as $N \to \infty$. This should be known, but I cannot find it in the literature.
- 2,207
9
votes
1
answer
458
views
Arguments of exponential sums
Let $p$ be a prime, let $\zeta_p=e^{2\pi i/p}$,
let $g\in{\bf F}_p$ be a non-square and
let $\chi:{\bf F}_p^*\rightarrow{\bf C}^*$ be a non-trivial character.
Then the complex numbers
$$
\chi(n)\...
- 670
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
9
votes
1
answer
204
views
Which unimodular lattices $L\subset \mathbb R^2$ minimize $f_t(L):=\sum_{ v\in L} e^{-t \|v\|_2}$? (for parameters $t>0$)
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}$Consider the lattices in $\SL(2,\mathbb R)(\mathbb Z^2)$ up to rotation. The space of such lattices can be identified with the modular surface $\...
- 241
9
votes
0
answers
232
views
Explicit bound for sum of Kloosterman sums
What are the best fully explicit upper bounds one can give for the sum
$$\left\lvert \sum_{n=N}^{\infty} \frac{S(a,b;n)}{n} \,I_1\!\left(\frac{4 \pi \sqrt{|ab|}}{n}\right) \right\lvert$$
where $S(a,b;...
- 2,235
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
$$\sum_{x=0}^{q-1} e_q(ax^2) = \left(\...
user40023
8
votes
1
answer
606
views
Riemann hypothesis for exponential sum
Recently I've heard about the Riemann hypothesis for one-variable exponential sums, which states as
For a polynomial $f\in\mathbb{F}_{p^k}[x]$ of degree $d$ and a character $\chi$ of $(\mathbb{F}_{p^...
- 181
8
votes
0
answers
418
views
Equidistribution of $\{\sqrt{p}: p \text{ primes }\}$ modulo 1
I am trying to show $\{\sqrt{p}: p \text{ primes }\}$ is equidistributed modulo 1. Using Weyl's criterion, it is sufficient to show for each nonzero integer $k$,
\begin{equation}
\sum_{n \leq x}e(k\...
- 91
8
votes
0
answers
398
views
$L^1$ norm of Fourier transform of subset sums
Let $n_1,\dots,n_k$ be a set of $k$ natural numbers less than $N$, with $k = (1- \delta) \log_2 N$ for $\delta$ relatively small. Let $e(x) = e^{ 2\pi i x}$, as usual.
Assume that $$\int_0^1\prod_{j=1}...
- 148k
7
votes
1
answer
518
views
Does there exist some irrational $x,\alpha$ so that this Weyl sum is $o(\sqrt N)$?
This is a less ambitious version of Is the Lebesgue measure of the $x$ so that this exponential sum is $o(\sqrt{N})$ positive? .
Consider $$S_N(x):=\sum_{n=1}^N \exp\left(2\pi i\left(\frac12n^2x+\...
- 1,904
7
votes
2
answers
531
views
Conjecture about an exponential sum
Let $X \subset \mathbb{N}$ and say that $X$ is super-equidistributed if
for all $\alpha \in \mathbb{R} \setminus \mathbb{Z}$ there exists $C(\alpha) > 0$ such that for all $N$
$$
\left| \sum_{x \in ...
- 334
7
votes
1
answer
321
views
optimal estimate for generalized Kloosterman sum
Let $p$ be an odd prime. Denote $e(x):=e^{2\pi i\frac{x}{p}}$. Let $n\ge 2$ be an integer. Consider the exponential sum
$$
S(f,g)=\sum_{g(x_1,\dots,x_n)=0, x_i\in\mathbb{F}_p}e(f(x_1,\dots,x_n)),
$$
...
- 103
7
votes
1
answer
656
views
Upper bound for an exponential sum involving characters of a finite field
Let $q = p^n $ be a prime power, $\alpha\in\mathbb{F}_{q} $
a primitive element of the finite field $\mathbb{F}_q$ and denote by $\chi $ a non-trivial additive character of $\mathbb{F}_{q} $. Set
$\...
- 93
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
6
votes
1
answer
283
views
Is the Lebesgue measure of the $x$ so that this exponential sum is $o(\sqrt{N})$ positive?
Consider $$S_N:=\sum_{n=1}^N \exp\left(2\pi i\left(\frac12n^2x+\alpha n\right)\right)$$
where $\alpha$ is irrational. For certain $x$ (say integer) we can get that this is bounded for all $N$. I am ...
- 1,904
6
votes
1
answer
287
views
Number of solutions for the inequality with square roots
Let $M$ be some large real number and $\delta>0$. I would like to estimate the number of solutions for the inequality
$$|\sqrt{n_1}+\sqrt{n_2}-\sqrt{n_3}-\sqrt{n_4}|<\delta\sqrt{M},$$
where $...
- 7,193
6
votes
1
answer
292
views
Prime number theorem via large sieve type sums
We know that the prime number theorem is equivalent to the statement
$$
M(x)=\sum_{n\le x}\mu(n)=o(x).
$$
By using Ramanujan sums, we can write $M(x)$ as
$$
M(x)=\sum_{q\le x}\sum_{\substack{0\lt a\le ...
- 178
6
votes
2
answers
646
views
Number of solutions of $am \equiv bn \pmod{q}$
Let $q$ be a (large) prime. Let $N$ be a positive integer of size${}\approx \sqrt{q}$. Let $\mathcal{M}$ be an arbitrary subset of $\{1, \dots, q\},$ such that $\mathcal{M}$ has cardinality $N$. ...
- 2,207
6
votes
2
answers
425
views
Average of gcd of sum of two $k$th powers
I am interested bounding the following quantity. Given fixed $k \in \mathbb{N}$, $a,b \in \mathbb{Z}$, $\sigma \in [0,1)$, and intervals $I_1,I_2 \subset \mathbb{Z}$ can we establish the bound
$$S = \...
6
votes
1
answer
183
views
Mean value of the divisor function over Piatetski-Shapiro sequences
Let $c>1$, $c\not\in\mathbb{Z}$ and consider the sum
$$
\sum_{n\leq x} \tau(\lfloor n^c \rfloor),
$$
where $\tau(n)$ is the number of divisors of $n$. I'm almost certain I've seen an evaluation of ...
- 1,990
6
votes
0
answers
108
views
Bounds on exponential and character sums of ratio of linear recurrences
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,...
- 121
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
5
votes
1
answer
405
views
Exponential sum involving floor function
Can one get cancellation in exponential sums such as, say,
$$
\sum_{n\sim N} e(\lfloor n^\theta\rfloor^\beta),
$$
for fixed positive $\theta,\beta\not\in\mathbb Z$? When $\theta < 1$, it seems ...
- 1,890
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:
$$\text{Jones}_K(e^{2\pi i/(k+2)})=\int_{\text{$SU(2)$ connections on $\mathbb S^3$}/\...
- 18.7k
5
votes
0
answers
104
views
Exponential sums with monomials with divisor-function coefficients
In their paper "Exponential Sums with Monomials," Fouvry and Iwaniec study exponential sums roughly of the form
$$
\sum_{m_1 \sim M_1} \cdots \sum_{m_r \sim M_r} c_1(m_1) \cdots c_r(m_r) e\...
- 1,990
5
votes
1
answer
629
views
How different can the bias of two polynomials be?
$\DeclareMathOperator\bias{bias}$I'm trying to figure out how to approach the following question:
Let $g$, $h$ be polynomials over $\mathbb{Z}_p$ (for prime $p$) with $n>1$ variables.
Denote by $\...
- 301
5
votes
0
answers
302
views
Exponential sums with prime power modulus
I am looking for an analogue of the following result of Fouvry and Katz for prime power modulus ("A general stratification theorem for exponential sums, and applications", J. reine angew. Math. 540 (...
- 26.9k
5
votes
0
answers
124
views
Linear exponential sum with gcd
The sum $$\sum _{d,d'\leq D}\sum _{h,h'=1}^q(h,q)e\left (\frac {dh+d'h'}{q}\right )$$ is easily seen to be $$\ll q^{2+\epsilon }+D^2.$$ Indeed with a standard estimate for a linear exponential sum it ...
- 143
5
votes
0
answers
370
views
Lower bound for some sums of roots of unity
Let $n$ be a positive integer (assume $n$ is prime for simplicity), and let $x_k = \pm1$, for $k = 0,1,2,..., n-1$. Let $\rho$ be an $n-$th primitive root of unity, I am interested in a lower bound ...
- 1,101
4
votes
1
answer
282
views
Mean square estimate for the Kloosterman sums
For $m,n\in \mathbb{N}$, denote the Kloosterman sum
$$S(m,n;c)=\sum_{a\bmod c}e\left( \frac{ma+n \overline{a}}{c}\right),$$where $\overline{a}$
denotes the multiplicative inverse of $a\bmod c$.
Does ...
- 353
4
votes
2
answers
332
views
estimate for a sum of products of Weil's sum
Let $p$ be a prime and consider the field $\mathbb{F}_p$. Fix $f\in\mathbb{F}_p[X]$ a polynomial of degree $d\ge 2$. Define
$$
K(x,y)=\frac{1}{\sqrt{p}}\sum_{z\in\mathbb{F}_p}e_p(xz+yf(z)),
$$
where $...
- 463
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
4
votes
1
answer
338
views
Bounds on quadratic character sums
I asked this question on Mathematics stack exchange but didn't get a response, so I ask here too.
Let $\chi$ be the non-trivial quadratic character of $\mathbb{F}_q$, and let $f(x)$ be a square-free ...
- 153
4
votes
1
answer
532
views
An Exponential Sum Restricted to Primes
Let $a,q,N$ be integers such that $N/2 \leq q \leq N$ and $a/q \notin \mathbb{Z}$.
Is the following estimate true, and, if so, how can it be proved?
\[\left|\sum_{1 \leq p \leq N} \exp(2\pi i p a/q) \...
- 217
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
4
votes
2
answers
366
views
On an observation which relates to the exponential sum $\sum_{n=1}^{[\sqrt{t/2\pi}]} n^{-\frac{1}{2}+it}$
This observation is based on the numerical calculation of the exponential sum:
$$\sum_{n=1}^{[\sqrt{t/2\pi}]} n^{-\frac{1}{2}+it}$$
It is known that this sum is related to the famous Riemann–Siegel ...
- 395
4
votes
1
answer
299
views
The Wilton-type bounds involving half-integral weight cusp forms
There is a basic question which puzzles me for a while, and maybe look naive for some experts here. The question is the following:
Let $f(z)=\sum_{n\ge 1} a_f(n) n^{k/2-1/4}e(nz)\in S_{k+1/2}(4N)$ be ...
- 563
4
votes
1
answer
332
views
Estimating certain short Kloosterman sums
Recall that for the classical Kloosterman sum
$$ K(a,b,p^t):= \sum_{x \in (\mathbb{Z}/ p^t \mathbb{Z})^* } \psi \left(\frac{ax+bx^{-1}}{p^t} \right),$$
where $\psi(x)=e^{2\pi ix}$, $a,b,t$ are natural ...
- 421