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.

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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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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'...
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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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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 ...
po0l's user avatar
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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 $\...
user84899's user avatar
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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 ...
Quicky2357's user avatar
3 votes
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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 $\tau_3(n) = \sum_{d_1d_2d_3=n} 1$ ...
Joshua Stucky's user avatar
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2 answers
753 views

Product of three or more independent sub-Gaussian varibles

A random variable $X$ is called subgaussian of order $\sigma^2$ if $\log E[exp\{\theta X\}]\leq \frac{1}{2}\theta^2\sigma^2$ for every $\theta\in\mathbb R$. Given a sequence of independent subgaussian ...
Tiago's user avatar
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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 ...
Lev Bahn's user avatar
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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 $n$ ...
user90189's user avatar
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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 ...
VSP's user avatar
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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 ...
Null_Space's user avatar
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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,...
Turbo's user avatar
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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 ...
MightyPower's user avatar
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2 answers
269 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 ...
joro's user avatar
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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 ...
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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^...
katago's user avatar
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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 ...
WhiteDwarf's user avatar
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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;...
Fredrik Johansson's user avatar
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1 answer
154 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 ...
GWB's user avatar
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How different can the bias of two polynomials be?

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 $bias(g)=|\sum_{x\in \mathbb{Z}_p^n}e^{...
GWB's user avatar
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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'...
Mini's user avatar
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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 $...
Shivin Srivastava's user avatar
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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 &...
Shivin Srivastava's user avatar
8 votes
1 answer
585 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^...
Jugendtraum's user avatar
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0 answers
408 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\...
Kyle Yip's user avatar
4 votes
1 answer
266 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 ...
Fei's user avatar
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2 votes
1 answer
274 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 \...
Iguana's user avatar
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7 votes
1 answer
644 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 $\...
nahila's user avatar
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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{...
shrinklemma's user avatar
4 votes
1 answer
313 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 ...
JACK's user avatar
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2 votes
0 answers
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Converse of Gallagher identity

A well known useful inequality of Gallagher states (in one form) that for any sequence $a:\mathbb N\to\mathbb C$, we have that $$\int_{|\theta|\le\delta} \bigg|\sum_n a(n)e(n\theta)\bigg|^2d\theta\ll \...
Mayank Pandey's user avatar
18 votes
2 answers
2k views

A finite alternating sum

We have stumbled upon the following finite alternating sum, which we have trouble analyzing. The sum is: $$ S_n = \sum_{j=0}^n \frac{ (-1)^j e^{-j} }{j!} (n-j)^j $$ We have observed numerically that ...
Francisco's user avatar
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An integral involving many exponential terms with quadratic exponents in the denominator

Given $k$ points $\{p_1,\cdots, p_k\}$ in $\mathbb{R}^n$ and positive constants $r_1, ..., r_k$ and another positive constant $\alpha>0$. Is there a way to compute/approximate the following (...
Min Wu's user avatar
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0 answers
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Way to express a number in its most compact sum of powers

Given a non-negative number n, what is the best approach to find the most compact representation for n in terms of sums of powers, such that the bases and the exponents can't surpass a given value (...
DaviFN's user avatar
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0 answers
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Formula for exponential integral over a cone

While reading 'Computing the Volume, Counting Integral points, and Exponential Sums' by A. Barvinok (1993), I came across the following: "Moreover, let $K$ be the conic hull of linearly independent ...
Erik's user avatar
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12 votes
2 answers
659 views

A question regarding Bourgain's paper on $\Lambda(p)$-subsets

I'm trying to understand Bourgain's proof of Proposition 1.10 on page 304-307 in On $\Lambda(p)$-subsets of squares which states Given $p>4$, we have the estimate \begin{align} \left\|\sum_{n=...
Jacky Chong's user avatar
3 votes
0 answers
126 views

A good way to bound the following exponential sum over $\mathbb{Z}/q\mathbb{Z}$ involving linear forms?

Let $q \in \mathbb{N}$. I am interested in getting an upper bound for the sum $$ \sum_{(a_1, a_2, a_3, q) = 1} \sum_{\mathbf{h} \in (\mathbb{Z}/q\mathbb{Z})^n }e( \frac{a_1}{q}\ell_1(h_1, \ldots, h_n)...
Johnny T.'s user avatar
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0 answers
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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 (...
Stanley Yao Xiao's user avatar
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 ...
caws's user avatar
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1 vote
1 answer
211 views

Upper bound for the integral over minor arcs of the exponential sum with prime omega function coefficients

Define $\mathfrak{m}$ as the union of the minor arcs of the form $|\alpha-\frac{a}{q}|\leq 1/qQ$, with $(a,q)=1$ and $Q_0<q\leq Q$, with $Q_0\geq N/Q$, for a certain $N\geq Q$ large. Is it ...
The Number Theorist's user avatar
4 votes
0 answers
161 views

Smoothed Weyl sum inequality

One version of Weyl's inequality states that for any $\alpha\in\mathbb{R}$ and $(a, q) = 1$ such that $|\alpha - a/q|\le 1/q^2$, we have that $$\sum_{n\le X} e(n^k\alpha)\ll X^{1 + \varepsilon}(q^{-1}...
Mayank Pandey's user avatar
7 votes
2 answers
507 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 ...
Lambert A'Campo's user avatar
3 votes
0 answers
91 views

Flow of zeros in the shifted exponential generating function?

Given a sequence $a_n$ (of real numbers, described more fully below), one may define the exponential generating function (on the complex plane) as $E(z)=\sum_{n=0}^\infty a_n z^n/n!$. The derivatives $...
Linas's user avatar
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5 votes
0 answers
333 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 ...
shurtados's user avatar
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1 vote
0 answers
53 views

Discrepancy bound of integer tensor product sequence?

Here discrepancy is from $(2.4)$ in https://www.ricam.oeaw.ac.at/files/people/siambook_nied.pdf given by 'The (extreme) discrepancy $D_N(P) = D_N(x_l,\dots,X_N)$ of the point set $P$ of $N$ points in $...
Turbo's user avatar
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3 votes
1 answer
230 views

Inequality for exponential sum in Dvoretzky 1972

I'm currently trying to figure out the following inequality. It looks like an inequality for the exponential sum, but I can't verify it or find a source explaining it any further. Most likely it has ...
DrShredz's user avatar
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3 votes
0 answers
199 views

Cancellation in this exponential sum?

I would like to know whether it is possible to obtain cancellation in the sum $$\sum_{p \leq X} e^{{2\pi iX}/{p}}$$ where $X$ is a real number that goes to $\infty$, and $p$ denotes a prime number.
Pablo's user avatar
  • 11.2k
0 votes
1 answer
265 views

The minimum of the maximum of a sequence of sinc functions

I apologise if this is trivial or well known to be impossible: Can one find a finite set of integers $2\leq a_1<a_2<\ldots<a_m<\infty$ such that for the function defined as $$ f_{a_1,\...
kodlu's user avatar
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1 vote
1 answer
497 views

Coefficients $U_m(n,k)$ in the identity $n^{2m+1}=\sum\limits_{0\leq k \leq m}(-1)^{m-k}U_m(n,k)\cdot n^k$

Review the main result of mathoverflow.net/questions/297900, that is the identity \begin{equation}\label{f1} n^{2m+1}=\sum\limits_{1\leq k \leq n}\sum\limits_{j\geq0}A_{m,j}k^j(n-k)^j, \end{equation} ...
Petro Kolosov's user avatar