Questions tagged [exponential-sums]
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 exponential sums is necessary for studying modern number theory.
196 questions
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On the upper-bound for a type of quintuple Kloosterman sums
Sorry to disturb, dear experts here. I have a question involving the quintuple Kloosterman sum, and expect some hints to show the upper-bound.
My question is, for any $x,y,z,w,\delta \in \mathbb{Z}$ ...
- 148k
3
votes
1
answer
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Estimates for certain double-Kloosterman sums
Sorry to disturb. I encounter a double-Kloosterman sum, which needs some help from the experts here.
For any $q\in \mathbb{N}^+$, how can we estimate the type of sum
$$ \sideset{_{}^{}}{^{\ast}_{}}\...
- 563
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answers
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Question about exponent pairs
In some of my recent research efforts, I've been applying a lot of estimates for exponential sums involving exponent pairs. Two seemingly simple questions have arisen from these calculations, and I ...
- 1,990
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A question involving the three-dimensional Kloosterman sum
Sorry to disturb. I have a question involving the three-dimensional Kloosterman sum, which needs some help from the experts here.
For any $\alpha, \beta, \gamma \in \mathbb{Z}$ and $q\in \mathbb{N}^+$,...
- 148k
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Exponential sums with monomials with divisor-function coefficients
In their paper "Exponential Sums with Monomials," Fouvry and Iwaniec study exponential sums roughly of the form
$$
\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
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Sums of Kloosterman sums
Let
\[ S_{n,m}(q)=\sum_{a=1\atop {(a,q)=1}}^qe\left (\frac {an+\overline am}{q}\right )\]
be Kloosterman's sum and $\alpha _n,\beta _m$ be complex numbe of modulus $\leq 1$. For $Q,N,M>0$ what is ...
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Self-referencing recurrence relation with exponential
I have the self-referencing recurrence relation
$$
d(0) = 0
$$
$$
d(1) = a
$$
$$
d(n+1) = d(n) + a*e^{-(\frac{d(n)-n*c}{b})^4}
$$
Written as a sum:
$$
d(N) = a*\sum_{n=0}^{N}e^{-(\frac{d(n)-n*c}{b})^...
- 11
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votes
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answer
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Bound for sum of multiplicative character calculated over multivariate polynomial
Let $f \in \mathbb{F}_q[x_1, \dots, x_k]$ be a polynomial with $\deg f = n$, and let $\chi$ be a multiplicative character over $\mathbb{F}_q$.
Is there any known bound, possibly with conditions about $...
- 148k
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Distribution of quadratic polynomials mod $n$ and $n^2$
Suppose $n$ is odd, then both equations $x^2 = D \; mod \;n$ and $x^2 = D \; mod \;n^2$ have the same number of solutions for fixed $D$ coprime to $n$. What can be said about the relationship between ...
CommunityBot
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Moments of an exponential sum
Let $p$ and $N$ be large natural numbers. I would like to get a possibly sharp asympotic approximation of
$$
\mathcal{I}_{p,N}=\int_0^1 \Big(\sum_{j=1}^N e^{2\pi i j \xi}+\sum_{j=1}^N e^{-2\pi i j \xi}...
- 421
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$L^1$ 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 $$\int_0^1\prod_{j=1}...
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On the $L^1$-norm of certain exponential sums
I am stuck with an elementary-looking problem, which does not belong to my usual field of research so I eventually decided to ask it on MO.
Let $S$ be a finite set of integers. For $P$ a subset of $S$,...
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Lower bound for exponential sums
Let $D$ be a subset of $\mathbb Z/n \mathbb Z$ containing $0$. For $m$ an integer, set $$\alpha(m,D)=\sum_{d \in D} e\left (\frac{m d }{n}\right ),$$
where as usual $e(x) = e^{2 i \pi x}$ This is an ...
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The exploration of the asymptotic behavior of a simple sum. $\sum_{k=1}^{\infty} (k^{1/k} - 1)$ [closed]
An informal investigation of a sum.
Consider this sum:
$$S =\sum_{k=2}^{\infty}(k^{1/k} -1)$$
Does this converge? How does it behave as it diverges, if it diverges? If $k$ equaled $1$ we would get $0$ ...
CommunityBot
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answer
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views
Why are exponential sums so bad at solving this very easy problem?
Exponential sums are a powerful tool in additive combinatorics and number theory. In my understanding, when it comes to estimate the cardinality of a certain set, exponential sums are (essentially) ...
- 4,862
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votes
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answer
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A conjecture relating an integral and a sum, the floor function and squares
I've found through evidence and have conjectured on a math publication that:
$$\Big\lfloor\int_1^\infty (k^{1/(k^{1+1/\sqrt{x}})} - 1)dk\Big\rfloor = \Big\lfloor\sum_{k=1}^{\infty}k^{1/(k^{1+1/\sqrt{x}...
CommunityBot
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Sum of Kloosterman sums with oscillating factor
Denote by $S(c;n,m)$ Kloosterman's sum. Take $X>0$ and take $n,m\in \mathbb Z$ smaller than a small power of $X$ in modulus. It is known that essentially
\[ \sum _{c\sim X}\frac {S(c;n,m)}{c}\ll ...
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Connection between cyclic group and exponential function
I have been thinking about this for a while, but now got to the point where I got stuck. I don't know if it might be considered as a research level question, but I would be very happy if somebody knew ...
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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
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votes
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answer
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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
$\...
CommunityBot
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votes
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answers
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growth rate of quadratic exponential sums with monomial coefficients
What is the growth rate of $$S_d(M)=\sum_{n=1}^M n^d e\left(\frac{n^2}{2M}\right)$$
where $M$ is an even integer.
My numerical experiments show that
$$\frac{S_d(M)}{M^{d+\frac{1}{2}}}\approx e^{\frac{\...
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votes
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answers
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What does this exponential sum evaluate to?
We have the following sum
$$S=\sum_{\substack{0<a'\leq k'\\(a',k')=1\\a'\equiv b \bmod q}}e(hl'a'/k').$$ Here, $e(x):=\exp(2\pi i x)$, $h,k',q,a'$ are all natural numbers. We do know that $\gcd(h,l'...
CommunityBot
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votes
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answers
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Generalizing an estimate of Jutila
I'm working on a problem right now in which I need an upper bound for an exponential sum of the form
$$
\tag{1}
\sum_{N < n \leq 2N} \tau_3(n) e(f(n)),
$$
where $\tau_3(n) = \sum_{d_1d_2d_3=n} 1$ ...
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answers
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Reference for a paper of Jutila
Does anyone know where I might be able to locate on the internet the following paper of Jutila?:
M. Jutila, Mean value estimates for exponential sums. Number Theory, Ulm 1987, 120-136.
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2
answers
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Exponential Series with a sequence [closed]
For a convergent sequence $(a_n)_n \rightarrow a$ consider the exponential series
\begin{equation*}
\exp_{(a_n)_n}(-x) := \sum_{n=0}^{\infty} \frac{(-x)^n a_n}{n!}.
\end{equation*}
Can there be ...
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votes
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answer
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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
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votes
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Upper bound an integral with exponential function
I am working on my research about approximation a function. I come up with the following integral. I run some simulations and saw that the integral would converge to zero as n goes to infinty. Here is ...
- 128k
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votes
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answer
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Deligne's theorem on exponential sums
I'm an analyst who needs to use Deligne's Theorem 8.4 in 1, but I feel lost in the maze of definitions, and I don't trust my geometric intuition here.
Theorem 8.4: Let $Q$ be a polynomial in $n$ ...
CommunityBot
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An inequality between sum of exponential functions wrt dyadic index
I am reading a paper 'Periodic Nonlinear Schrodinger Equation and Invariant Measures' written by J.Bourgain. And I am wondering if I can have some help from this website.
My question is an inequality ...
- 128k
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votes
1
answer
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Maximum number of positive roots is $3$
Let $$f(x) = a+b(x+p)^t+c(x+p)^t(x+q)^t+d(x+p)^t(x+q)^t(x+r)^t,$$
where $t>1$ is any positive real number, $p>q>r>0$ or $p<q<r$ are positive integers and $a,b,c,d$ are any ...
- 128k
0
votes
1
answer
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Solving the inequality between a and b [closed]
I run into this inequality
$$
(a + b)^{1 - \epsilon} \;a < b
$$
where $a \in \mathbb{Z}^+$ and $\epsilon \in (0, 1)$. What value (w.r.t $a$ and $\epsilon$) should I set $b$ equal to such that this ...
- 128k
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vote
0
answers
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views
Polynomial composition utilizing polynomials in two different finite fields
At every $n\in\mathbb N$ (all polynomials are of degree $O(1)$) is there $g_{3,1}^{(n)},\dots,g_{3,k}^{(n)}\in\mathbb F_3[x_1,\dots,x_n]$ at $k=\mathsf{poly}(n)$ and $g_2^{(n)}\in\mathbb F_2[x_1,\dots,...
- 13.9k
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votes
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answer
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Twisting by a multiplicative Character in Katz, Perversity and Exponential sums
Let $C(x_1,\ldots,x_n)$ be a nonsigular cubic form with integral coefficients.
In his proof that $C$ fulfills the Hasse-Principle, if $n\geq 9$, Hooley used the following estimate that was provided by ...
- 8,842
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votes
1
answer
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views
Can the following sum be counted or expressed in terms of special functions?
Let us define this sum as a function of $z \in \mathbb{C}$ with some positive parameter $a$
$$
f(z; a) = \sum\limits_{n = 0}^{\infty}\frac{|z|^{2n}}{n!}e^{-ian^2}.
$$
Probably, it can be expressed (or ...
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votes
2
answers
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views
Closed form for $\sum_{i=1}^n{a^{i^2}}$
Let $a$ be an element of some ring or field, possibly finite.
Is there closed form for $\sum_{i=1}^n{a^{i^2}}$?
sage and wolframalpha couldn't solve it.
If $a$ is primitive n-th root of unity this is ...
- 25.4k
1
vote
0
answers
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views
What is the bound for $L_{\infty}$ norm for positive part of exponential sum?
A famous conjecture posed by Littlewood and solved by O. McGehee, L. Pigno, and B. Smith(in their article, Hardy's inequality and the $L_1$-norm of
exponential sums) and S. V. Konyagin independently ...
- 543
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answers
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views
Is this limit zero?
Define $e(\theta)=e^{2\pi i\theta}, \theta\in [0,1]$, $P_n=\{p_1,p_2,...,p_n\}$ are the first $n$ primes, $\|f\|_1=\int_{[0,1]}|f(\theta)|d\theta$.
Problem 1.
is it true for all fixed $m\in \mathbb{N^...
- 188k
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Explicit bound for sum of Kloosterman sums
What are the best fully explicit upper bounds one can give for the sum
$$\left\lvert \sum_{n=N}^{\infty} \frac{S(a,b;n)}{n} \,I_1\!\left(\frac{4 \pi \sqrt{|ab|}}{n}\right) \right\lvert$$
where $S(a,b;...
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votes
1
answer
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views
An asymptotic expansion of a infinite sum
I am interested in the asymptotic expansion in $t$($t>0$) when $t\to 0^+$ of the following series
$$
\sum_{k\ge 0}e^{-k^{2/n}t}
$$
for integer $n>2$ (n=1 follows from Poisson summation formula ...
- 128k
3
votes
1
answer
171
views
Exponential sums over rings
I'm trying to evaluate an exponential sum of the form:
\begin{equation}
\sum_{c\in Z_q}\chi(f(c))
\end{equation}
For polynomial $f(x)=a_2x^2+a_1x+a_0$ (with $a_2\ne 0$).
If $q$ is prime, then this is ...
- 10.4k
1
vote
0
answers
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views
Growth rate of exponential sum of $S_j$
Let $X_j$, $1 \leq j \leq n$ be chosen i.i.d. uniform over $[0,2\pi)$. Denote $S_j \triangleq X_1 +X_2+\cdots+X_j$ and suppose that $c_j$, $1 \leq j \leq n$ are some constants such that $|c_j|=1$.
I'...
- 85
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vote
0
answers
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views
Find conditions for the following running average to be monotonically decreasing
Let $S_n$ be defined as $\frac{1}{n}\sum_{t=1}^{t=n} [px_t^2 - (p+q)x_t]$ where $x_t = 1-(1-p-q)^t$. We want to find the conditions on $p$ and $q$ such that $S_n$ is monotonically decreasing for all $...
0
votes
1
answer
134
views
Prove that the following running average is monotonically decreasing
Let $S_n$ be defined as $\frac{1}{n}\sum_{t=1}^{t=n} [x^2+(p-q)x]$ where $x = 1-(1-p-q)^t$. We want to find the conditions on $p$ and $q$ such that $S_n$ is monotonically decreasing for all $n$. $0 &...
- 947
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votes
2
answers
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views
Hypotheses for exponent pairs
The theory of exponent pairs provides bounds for $$\sum_{N<n<2N} e(f(n)),$$ where f behaves like a monomial. Precise formulations of this are in Graham and Kolesnik (GK) which seems to be what ...
8
votes
1
answer
606
views
Riemann hypothesis for exponential sum
Recently I've heard about the Riemann hypothesis for one-variable exponential sums, which states as
For a polynomial $f\in\mathbb{F}_{p^k}[x]$ of degree $d$ and a character $\chi$ of $(\mathbb{F}_{p^...
- 148k
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votes
0
answers
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views
Equidistribution of $\{\sqrt{p}: p \text{ primes }\}$ modulo 1
I am trying to show $\{\sqrt{p}: p \text{ primes }\}$ is equidistributed modulo 1. Using Weyl's criterion, it is sufficient to show for each nonzero integer $k$,
\begin{equation}
\sum_{n \leq x}e(k\...
- 91
4
votes
1
answer
282
views
Mean square estimate for the Kloosterman sums
For $m,n\in \mathbb{N}$, denote the Kloosterman sum
$$S(m,n;c)=\sum_{a\bmod c}e\left( \frac{ma+n \overline{a}}{c}\right),$$where $\overline{a}$
denotes the multiplicative inverse of $a\bmod c$.
Does ...
- 43.7k
2
votes
1
answer
316
views
Integral over an exponential sum with squares
How should I estimate the following integral
$$I = \int_0^1 \left( \sum_{n=0}^{p-1} e(n^2t) \right)^2 dt $$
where $p$ is a prime?
Here is the method I followed:
\begin{align*}
I & = \int_0^1 \...
- 7,013
0
votes
0
answers
221
views
Number of integer solutions to quadratic polynomial with integer coefficients
It is known from for example Representations of Integers as Sums of Squares by Grosswald, E. that
$$|\{(n_1,n_2,\ldots,n_k)\in\mathbb{Z}^k: \ n_1^2+n_2^2+\cdots+n_k^2=N\}|\leq C_\varepsilon N^{\frac{...
- 193
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votes
2
answers
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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 ...
- 63.9k