Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [exponential-sum]

For questions on exponential sums, such as $\sum \exp(2\pi ix_n)$.

In mathematics, an exponential sum may be a finite Fourier series (i.e. a trigonometric polynomial), or other finite sum formed using the exponential function, usually expressed by means of the function $ \exp(2\pi ix).\,$ Therefore a typical exponential sum may take the form $$ \sum \exp(2\pi ix_n), $$ summed over a finite sequence of real numbers $x_n$.

496 questions
60
votes
0 answers

A Nice Problem In Additive Number Theory

$\color{red}{\mathbf{Problem\!:}}$ $n\geq3$ is a given positive integer, and $a_1 ,a_2, a_3, \ldots ,a_n$ are all given integers that aren't multiples of $n$ and $a_1 + \cdots + a_n$ is also not a multiple of $n$. Prove there are at least $n$…
James Moriarty
  • 6,460
  • 1
  • 20
  • 39
28
votes
1 answer

Show that $ \lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{k=0}^{n-1}e^{ik^2}=0$

TL;DR : The question is how do I show that $\displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}e^{ik^2}=0$ ? More generaly the question would be : given an increasing sequence of integers $(u_k)$ and an irrational number $\alpha$, how do I…
Renart
  • 3,046
22
votes
3 answers

Baker-Hausdorff Lemma from Sakurai's book

I'd like to show that, given to hermitian operators $A,G$ on a Hilbert space $\mathscr{H}$, the following identity holds: $$ e^{iG\lambda}A e^{-iG\lambda} = A + i\lambda [G,A] +…
Brightsun
  • 6,963
22
votes
3 answers

log of summation expression

I am curious about simplifying the following expression: $$\log \left(\sum_\limits{i=0}^{n}x_i \right)$$ Is there any rule to simplify a summation inside the log?
KennyYang
  • 429
18
votes
1 answer

Number of zeros in difference of exponential sums: $\sum\limits_{i=1}^n a_i^x - \sum\limits_{i=1}^n b_i^x$

Let $$f(x) = \sum\limits_{i=1}^n a_i^x - \sum\limits_{i=1}^n b_i^x$$ where the $a_i$ and $b_i$ are positive reals such that $f(x)$ is not a constant zero for all real $x$. Is it possible to find a maximum possible number of zeros of $f(x)$ and how…
Henry
  • 169,616
10
votes
2 answers

Find all positive integers $a, b, c$ such that $21^a+ 28^b= 35^c$ .

Find all positive integers $a, b, c$ such that $21^a+ 28^b= 35^c$. It is clear that the equation can be rewritten as follows: $$ (3 \times 7)^a+(4 \times 7)^b=(5 \times 7)^c $$ If $a=b=c=2$ then this is the first possible answer to this issue.…
Anatoly Karpov
  • 105
9
votes
4 answers

Number of zeros in polynomial-exponential sums

Is there some bound (or even an exact solution) on number of real roots of polynomial-exponential sum of type $$f(x) = a_1b_1^x+a_2b_2^x+\cdots=\sum_{i=1}^N a_i b_i^x = 0$$ where $b_i>0, a_i\in\mathbb{R}$. Clearly, if $N=1$, there is no root ($a_1…
pisoir
  • 1,669
  • 1
  • 19
  • 32
9
votes
0 answers

Why should $\sum_{m=1}^N e(\alpha m^3)$ be big for some $\alpha?$

I'm going through a "circle method" proof of the fact that every large enough natural number $n$ is the sum of nine cubes. At some point a lot of control over the function $$f(\alpha)=\sum_{m=1}^N e(\alpha m^3)$$ is needed. Here $N=\lfloor…
augustoperez
  • 3,326
  • 12
  • 26
9
votes
3 answers

Prove the following series $\sum\limits_{s=0}^\infty \frac{1}{(sn)!}$

Prove that, $$\sum\limits_{s=0}^\infty \frac{1}{(sn)!}=\frac{1}{n}\sum\limits_{r=0}^{n-1}\exp\left(\cos\frac{2r\pi}{n}\right)\cos\left(\sin\frac{2r\pi}{n}\right)$$ I don't have a real idea on how to start approaching this question, some hints and…
user51515
  • 121
  • 1
  • 7
9
votes
0 answers

Equidistribution of roots of prime cyclotomic polynomials to prime moduli

Here is a relevant - and longstanding, I'm told - conjecture. Let $f \in \mathbb{Z}[x]$ be irreducible and of degree > 1. Set $E_p = \{x/p \: | \: 0 \leq x < p, f(x) \equiv 0 \: (p) \}$ = { normalised least positive residues of zeros of $f$ in…
Joshua Seaton
  • 2,258
7
votes
3 answers

Find all real solutions of $6^x+1=8^x-27^{x-1}$

Find all real solutions of $6^x+1=8^x-27^{x-1}$. Things I tried: We want solutions of $$2^x3^x+1 = (2^x)^3-\frac{(3^x)^3}{27}.$$ Write $a=2^x$ and $b=3^x$. This gives $$ab+1 = a^3-\frac{b^3}{27}$$ or $$27ab+27=27a^3-b^3$$ How to continue?
Jolien
  • 1,695
7
votes
1 answer

A nested double sum(to do with e?)

I’ve come across this nested double sum while doing an investigation but cannot seem to find a closed form for it. $$1+\sum_{i=1}^\infty{\frac{1}{i!} \sum_{j=0}^i}{\frac{1}{j!}}$$ This is about the simplest form I can get it to. I’m pretty sure it…
Habeeb M
  • 337
  • 1
  • 10
7
votes
1 answer

For which $r \in \mathbb R$ is the series $S(r)$ finite?

For each $r \in \mathbb R$ we let $$L_r := \left\{ \begin{pmatrix} a+cr \\ b+dr \\ c \\d \end{pmatrix} : a,b,c,d \in \mathbb Z \right\}, \quad W:= \left\{ \begin{pmatrix} 0 \\ 0 \\ x_3 \\ x_4 \end{pmatrix} : x_3,x_4 \in \mathbb R \right\} $$ and $$…
principal-ideal-domain
  • 6,324
7
votes
1 answer

A curious property of exponential sums for rational polynomials?

An article led me to generate some graphs of exponential sums of the form $S(N)=\sum_{n=0}^Ne^{2\pi i f(n)}$, where $f(n)= {n\over a}+{n^2\over b}+{n^3\over c}$ with $a,b,c\in\mathbb{N}_{>0},\,$ leaving me amazed at their great variety. Here are…
r.e.s.
  • 15,537
7
votes
2 answers

Number of zeros of a weighted sum of exponentials

Let be $n$ an integer. Let be $a_1, \ldots, a_n \in \mathbb{R}$ not all null and $b_1 < \ldots < b_n$ reals. Let be $f : x \mapsto \sum\limits_{i=1}^{n} a_i \exp(b_i x)$. I am trying to show that $f$ can be null over at most $n - 1$ points. What I…
Raito
  • 1,940
  • 11
  • 17
1
2 3
…
33 34