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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2
votes
1answer
103 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 \...
2
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0answers
53 views

$L^p$ bound on a Weyl sum

Consider the Weyl sum $$S(x,t)=\sum_{n=1}^Ne(nx+nt^2).$$ We have the Strichartz estimate $$\left\|S\right\|_{L^p(\mathbb{T}^2)}\lesssim N^{1-\frac{3}{p}},\ \text{for all }p>6.$$ We also have for $...
6
votes
1answer
346 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 $\...
0
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0answers
136 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{...
4
votes
1answer
225 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 ...
2
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0answers
66 views

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 \...
18
votes
2answers
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 ...
1
vote
0answers
131 views

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 (...
1
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0answers
172 views

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 (...
1
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0answers
57 views

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 ...
10
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2answers
324 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=...
3
votes
0answers
116 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)...
5
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0answers
218 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 (...
5
votes
0answers
97 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 ...
1
vote
1answer
110 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 ...
4
votes
0answers
98 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}...
7
votes
2answers
390 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 ...
3
votes
0answers
74 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 $...
5
votes
0answers
199 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
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0answers
44 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 $...
3
votes
1answer
199 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 ...
3
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0answers
156 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.
0
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1answer
84 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,\...
1
vote
1answer
365 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} ...
4
votes
1answer
275 views

Explicit numbers with square root cancellation in Weyl's exponential sum

I'm interested in examples of real numbers $\alpha$ where we have $$\left| \sum_{n=1}^N \mathrm e(\alpha n) \right| \ll N^{1/2} $$ or perhaps with the weaker estimate with the right side replaced ...
6
votes
3answers
538 views

Summing the infinite series $\sum_{k=0}^{\infty} \frac{x^k}{(k!)^2}$ [closed]

Is there a closed form sum of $\sum_{k=0}^{\infty} \frac{x^k}{(k!)^2}$ It is trivial to show that it is less than $e^x$ but is there a tighter bound? Thanks
3
votes
2answers
294 views

Proof of the identity $\left\lvert\sum_{j=0}^{p-1}e^{(2\pi i/p)(mj^2+nj)}\right\rvert=\sqrt p$ for $p$ odd prime

While studying themes related to mutually unbiased bases, I've come across the following identity: $$\left\lvert\sum_{j=0}^{p-1}e^{(2\pi i/p)(mj^2+nj)}\right\rvert=\sqrt p,$$ for $p$ odd prime and $m\...
2
votes
1answer
106 views

Uniform power-saving estimate for an exponential sum

Let $N$ be a large natural number. Define an expoential sum $$ I_m=\sum_{x,y=1}^N e^{2\pi i\frac{x^2-y^2}{N}m}, \,\, m=1,2,...,N-1. $$ The trivial bound for $I_m$ is $N^2$, as there are $N^2$ terms. ...
1
vote
1answer
604 views

Coefficients in the sum $\sum_{k=0}^{n-1}\sum_{j=0}^{m}A_{j,m}(n-k)^jk^j=n^{2m+1}, \ m=1,2,…$

Consider a sum $$\sum_{k=0}^{n-1}\sum_{j=0}^{m}A_{j,m}(n-k)^jk^j$$ which returns an odd power $n^{2m+1}$ of $n$, for $\ m=0,1,2,...$ given fixed $A_{0,m}, \ A_{1,m}, \ ..., \ A_{m,m}$. The ...
3
votes
0answers
102 views

Approximating $1_I$, $I\subset \lbrack 0,1\rbrack$, by trigonometric polynomials

Let $I$ be a subinterval of $\lbrack 0,1\rbrack$. How well can one hope to approximate it in the $L_1$-norm (on $\mathbb{R}/\mathbb{Z}$) by a trigonometric polynomial of degree $\leq R$, that is, a ...
2
votes
1answer
165 views

Moments of certain exponential sum

Let $v(\beta) := \sum_{n\le X} e(n\beta)$ where $e(\alpha) := e^{2\pi i \alpha}$. It is not hard to show that $$\log X\ll \int_0^1 |v(\beta)| d\beta\ll \log X$$ and by considering the underlying ...
5
votes
1answer
190 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 $...
3
votes
0answers
217 views

Prerequisites to read Katz's Gauss Sums, Kloosterman Sums, and Monodromy Groups

What is the minimal background one must have to read Katz's Gauss Sums, Kloosterman Sums, and Monodromy Groups?
3
votes
2answers
265 views

How to estimate a mixed character sum $\sum_{h \in \mathbb{Z}/q \mathbb{Z}} \chi(f(h)) e(Ch/q)$?

Let $q = p^t$ where $p$ is prime. I am interested in estimating the complete exponential sum, which looks like $$ \sum_{0 \leq h < q} \chi( (h-a_1)(h-a_2)(h-a_3)) \ \bar{\chi}( (h-b_1)(h-b_2)(h-...
2
votes
0answers
176 views

An exponential sum like the Kloosterman sums

I encounter a tricky sum like the Kloosterman sum $${\sum_{x \mod P}}^\ast e\left(\frac{ax+\overline{x}}{P}+\frac{lx^2}{P^2}\right),$$ where $l$ is a positive integer co-prime with $P$ and here $P$ ...
5
votes
1answer
242 views

Does a Kloosterman sum composed with a rational function exhibit square root cancellation?

Denote the classical Kloosterman and Salié sums, respectively, as $KL(a,b) = \sum_{r \in F_*} e(ar+\frac{b}{r})$ and $SL(a,b) =\sum_{r \in F_*} \chi(r) e(ar+\frac{b}{r})$, where $\chi(\cdot)$ is the ...
-4
votes
1answer
227 views

Does $\int_{0}^{\infty}e^{-xz}\sum_{n=0}^{\infty}a_{n}\frac{x^n}{n!}dx$ converge for $z>0$ with $a_{n} > n! $, for $ n>1$? [closed]

Let $g$ be exponential generating function such that $g(x)= \sum_{n=0}^{+\infty}a_{n}\frac{x^n}{n!}$ extended by analytic continuation along $\mathbb{R+}$ and has a positive radius of convergence. We ...
1
vote
0answers
106 views

Exponential sum over polynomial values

Let $f(x)=a_{k}x^{k}+\dots+a_{1}x+a_{0}\in \mathbb{R}[x]$ with $a_{k}>0$ and $N\in\mathbb{N}$ sufficiently large. I would like to know an estimate of the following sum: $$\sum_{N\leq n\leq 2N}\exp(...
1
vote
0answers
109 views

How to evaluate this sum of roots of unity with condition to zero

In evaluating the sum: $$\tag{1}\label{e1}\sum\limits_{j = 1 < h}^N {\left( {{e^{\frac{{i\pi }}{N}\left( {{s_1}j + {s_2}h} \right)}} - {e^{\frac{{i\pi }}{N}\left( {{s_1}h + {s_2}j} \right)}}} \...
4
votes
2answers
345 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 ...
1
vote
0answers
121 views

Is this averaged exponential sum over primes small infinitely often?

Do there exist infinitely many positive integers $N$ such that $$\sum_{\substack{N/2 \leq q \leq N \\ a/q \notin \mathbb{Z}}} \left|\sum_{1 \leq p \leq N} \exp(2\pi i p a/q) \right|\leq |a|^{o(1)} N^...
2
votes
0answers
98 views

An exponential sum estimate on small intervals

Let $1<r<2$ be a real number. Let $4<p\le 6$. Consider the exponential sum estimate $$\int_0^{2\pi}\int_0^{N^{r-2}} \left|\sum_{n=1}^N e^{inx+in^2 y}\right|^p \, dy \, dx$$ Notice that the $y$...
1
vote
0answers
78 views

Upper bound for $ \sum_{p\leq x} a(p)p^{it} $

The Vinogradov-Korobov zero-free region for the Riemann Zeta-function would give us an upper bound for $ \sum_{p\leq x} p^{it} $. Now my question is, what would be the corresponding upper bound for ...
0
votes
0answers
146 views

Bounding exponential sum of the form $\sum_{\mathbf{x} \in (\mathbb{Z}/q \mathbb{Z})^n } \chi_1(x_1)\cdots \chi_n (x_n) e(a F(\mathbf{x})/q)$

I have encountered the following exponential sum and I would like to obtain a non-trivial upper bound for it. I am not quite sure where to start, and I would greatly appreciate any suggestions on how ...
2
votes
1answer
162 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 ...
7
votes
1answer
212 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 $$ S(f,g)=\sum_{g(x_1,\dots,x_n)=0, x_i\in\mathbb{F}_p}e(f(x_1,\dots,x_n)), $$ ...
4
votes
2answers
296 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 $...
11
votes
1answer
513 views

exponential sum over variety

I am wondering where to find a good reference for bounds of the type $$\sum_{x\in V(\mathbb{F}_p)} \chi(g(x))\psi(f(x))$$ where $V$ is a variety, $\chi$ is a multiplicative character over $\mathbb{...
9
votes
1answer
337 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)\...
-4
votes
1answer
317 views

How many integers between $\left[2^{2^k}, 2^{2^{k+1}}\right]$? [closed]

Suppose $k$ and $n$ are natural numbers such that $2^{2^k} \lt n \lt 2^{2^{k+1}}$. I am curious how many integers are there in the interval $\left[2^{2^k}, 2^{2^{k+1}}\right]$ in terms of $n$. I need ...