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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.

8
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1answer
214 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
54 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 $...
4
votes
0answers
154 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
34 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 $...
0
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0answers
102 views

Difference between Dirichlet Pigeonhole and Exponential sums bound in particular situation?

Dirichlet Pigeonhole says given prime $p$ and vector $(v_1,v_2,\dots,v_n)\in\mathbb Z^n$ there is an integer $m$ such that the vector $m(v_1,v_2,\dots,v_n)\bmod p$ lies in box $[-p^{(n-1)/n}-1,p^{(n-1)...
3
votes
1answer
188 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
132 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
70 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
324 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
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1answer
225 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
516 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
273 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
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1answer
88 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
415 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 ...
0
votes
0answers
54 views

Closed Form or Special Function Expression for Basic Looking Geometric-Like Sum

Is there any known closed form, involving any commonly used special function, of sums of the type $$\sum_{k=0}^{n-1} a^{k^m}$$ for m = 2, 3, ... It can be generalized to $m$ being a general complex ...
0
votes
0answers
85 views

On existence of certain primes and integers

Given $N\gg 0$ and small $\epsilon>0$ take coprime $A,B\approx N^{\frac14+\epsilon}$. From exponential sums we can show that for every prime $p\approx N$ (but $>N$) there is an $m\in\Bbb Z$ such ...
3
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0answers
95 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
146 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
171 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 $...
0
votes
0answers
147 views

Could a full rank linear system arise from this construction?

Fix a small ${\epsilon}>0$ and take a very large $n>0$. We say integers $A,B,C,D,A',B',C',D'$ satisfy property $P[n]$ if $A,B$ is coprime, $C,D$ is coprime, $A',B'$ is coprime and $C',D'$ is ...
3
votes
0answers
187 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
218 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
126 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
201 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
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1answer
218 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 ...
0
votes
0answers
125 views

Jacobi sum with half integers

Given that the following identity $\sum_{n=1}^{\infty}e^{-\pi n^2 a}=(\frac{1}{2\sqrt{a}}-\frac{1}{2})+\frac{1}{\sqrt{a}}\sum_{n=1}^{\infty}e^{-\pi n^2 /a}, a>0$ which can be proved using Poission ...
0
votes
0answers
84 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
94 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
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2answers
341 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
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0answers
108 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
94 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
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0answers
73 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
128 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
130 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
183 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
277 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 $...
9
votes
1answer
426 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{...
0
votes
0answers
705 views

On a number theoretic problem

Motivation: This problem comes from attempts to build a decoder for On a number theoretic problem coming from multiuser coding? and other related ramifications. Are there integers $a,b,|U|,|V|$, $N$...
9
votes
1answer
318 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
113 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 ...
0
votes
1answer
119 views

Counting solutions of a certain diophantine equation

For some $s, k$, let $J_{s, k}(X; \mathbf{n})$ be the number of solutions to the system $$\sum_{i\le s} (x_i^j - y_i^j) = n_j$$ for $j\le k$ with $x_1, \dots, x_s, y_1, \dots, y_s\in [1, X]\cap\mathbb{...
1
vote
0answers
98 views

Convergence of a tetration series

Let $0<a\neq 1$ be a fixed real number and denote by $a^{ \frac{x}{}}:=\mbox{uxp}_a(x)$ the ultra exponential function (ultra power) that is a unique extension of tetration (and also its linear ...
1
vote
0answers
72 views

Obtaining a “one-term” lower bound from an exponential sum lower bound of Roth

In the excellent book "A Panorama of Discrepancy Theory", edited by Chen et al, there is a chapter entitled Multicolor Discrepancy of Arithmetic Structures, by Hebbinghaus and Srivastav. From p. 360: ...
2
votes
1answer
107 views

Upper bound for an exponential sum in Waring-Goldbach problem

In Waring's problem, we have Hua's estimate $$S(a,b,q) = \sum_{x=1}^q e^{2\pi i (ax^k + bx)/q)} \ll q^{1/2+\epsilon} \gcd(b,q),$$ where $(a,q)=1$. ?Do you know a similar upper bound for the sum $$...
3
votes
0answers
283 views

On discrepancy of integer sequences related to Erdos-Turan-Koksma

Assume we have a sequence $a_1,\dots,a_k\in\Bbb N$ with each $a_i\approx n$ where $n$ is some integer. Suppose there exists an $\eta\in(0,1)$ such that for every $d_1,\dots,d_k\in\Bbb Z$ with $$...
-2
votes
1answer
193 views

Solutions to a diophantine system

What is the smallest $\gamma_1,\gamma_2,\gamma_3>0$ such that given coprime $p,q=\Theta(\ell)$ and integer $t\geq3$ there are coprime $m,n=\Theta(\ell^{t-1})$ with $(mn,pq)=1$, $\alpha_i\in\Bbb Z$ ...
2
votes
1answer
146 views

Upper bound for a higher dimensional Ramanujan sum

Fix an integer vector $\mathbf m\in \mathbb Z^k$. Let $q$ be a positive integer. Is there a "good" upper bound in terms of $q,\bf m$ for the exponential sum: \[\sum_{\mathbf n} e\left(\frac{\langle m,...
7
votes
0answers
208 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=...
2
votes
2answers
692 views

Occurrence of simultaneous small remainders?

Fix $(a,b)=1$, $a<b<2a$ and $a,b>n^{1/(2t)}$ and fix a prime $T\approx n^{\tau+\frac1k}$ where $\tau\geq1$ and $k=2(t-1)$. We can show using exponential sums there is an $m_{_T}$ such that $T/...
4
votes
1answer
313 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) \...