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

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9
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0answers
215 views

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 $E_p = \{x/p \: | \: 0 \leq x < p, f(x) \equiv 0 \: (p) \}$ = { ...
5
votes
1answer
151 views

Simplify exponential sum over $\mathbb{F}_p$ to prove identity

I have a sum involving $p$-th roots of unity (where $\frac{1}{t}$ is to be understood as the field inverse $t^{-1} \bmod p$ etc.) of the form $\begin{align*} &d_{j,k}=\sum_{a,b,c \in ...
5
votes
1answer
387 views

Integrating $\sum_{n=1}^{\infty} \frac{\sin nx }{n^4}$

Consider : $\displaystyle f(x)= \sum_{n=1}^{\infty} \frac{\sin nx }{n^4}$ Find : $\displaystyle \int_0^{x} f(t)\ \mathrm{d}t$.
4
votes
3answers
125 views

Sum of the sequence

What is the sum of the following sequence $$\begin{align*} (2^1 - 1) &+ \Big((2^1 - 1) + (2^2 - 1)\Big)\\ &+ \Big((2^1 - 1) + (2^2 - 1) + (2^3 - 1) \Big)+\ldots\\ &+\Big( (2^1 - 1)+(2^2 - ...
4
votes
1answer
202 views

Determining the Value of a Gauss Sum.

Can we evaluate the exact form of $$g\left(k,n\right)=\sum_{r=0}^{n-1}\exp\left(2\pi i\frac{r^{2}k}{n}\right) $$ for general $k$ and $n$? For $k=1$, on MathWorld we have that ...
4
votes
2answers
528 views

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] + ...
4
votes
2answers
687 views

upper bound of exponential function

I am looking for a tight upper bound of exponential function (or sum of exponential functions): $e^x<f(x)$ when $x<0$ or $\sum_{i=1}^n e^{x_i} < g(x_1,...,x_n)$ when $x_i<0$ Thanks a ...
4
votes
1answer
67 views

Character Sums, Weights, and Cohomology

I know, vaguely, that certain bounds for character sums over finite fields can be determined by looking at weights and the dimensions of certain compactly supported cohomology groups. However, I've ...
4
votes
0answers
99 views

Multiplicative subgroup of a finite field with prescribed trace.

Any suggestions/methods/estimates for the following problem would be very appreciated. $l,p$ are primes with $p \equiv 1 \!\! \pmod l$. $r$ is a positive integer with $r \equiv 1 \!\! \pmod p$ and $q ...
3
votes
3answers
68 views

How does $\displaystyle \frac{e^{-x}}{1-e^{-x}}$ become $\sum_{k=1}^{\infty}e^{-kx}$?

How does $\displaystyle \frac{e^{-x}}{1-e^{-x}}$ become $\sum_{k=1}^{\infty}e^{-kx}$? I know $\displaystyle \frac{e^{-x}}{1-e^{-x}}=\frac{1}{e^x-1}=\left(\sum_{k=0}^{\infty} ...
3
votes
2answers
110 views

How can I resolve $\sum_{x=0}^{\infty} xe^{-x/\theta}$? [duplicate]

I stumble on this summation during an exercise. How can I resolve $\sum_{x=0}^{\infty} xe^{-x/\theta}$?
3
votes
1answer
406 views

How to find a summation of a logarithmic function?

Suppose that I had to find $\log_{10}(8952!)$. Now, since $\log(a) + \log(b) = \log(ab)$, this can be rewritten to the following summation: $$\sum_{x=1}^{8952}{\log_{10}(x)}$$ Would there be a ...
3
votes
2answers
98 views

Sum of finite series of $\exp(an^2 + bn + c))$

Is there a way to simplify that sum to an expression without actual performing the summation, similar to the formula for calculating the sum of a (finite) geometric series? $\sum_{n=0}^{N-1} ...
3
votes
3answers
235 views

Summation evaluation of $\sum_{j=0}^k n^{1/2^j}$?

How do I go about solving this: $\sum_{j=0}^k n^{1/2^j}$ So, the terms of this series are $n , n^{1/2},n^{1/4},n^{1/8},.......n^{1/2^k}$ Any insights on what the thought process should be, to ...
3
votes
1answer
345 views

Proof for generalized sum of powers

Bernouli's Formula for sum of kth powers of first n natural numbers is given by: $$f_k(n)=\frac{1}{k+1}\sum_{j=0}^k{k+1\choose j}B_j(n+1)^{k+1-j}$$ where $Bj$ is the $j^{th}$ Bernoulli Number and is ...
3
votes
2answers
88 views

Inequality with a sum

I am reading Remarks on a Ramsey theory for trees: Janos Pach, Jozsef Solymosi, Gabor Tardos http://arxiv.org/abs/1107.5301 I am stuck at inequality in proof of Lemma 6. $n\geq 8$, $k=2\lfloor ...
2
votes
3answers
84 views

Exponential equation in x,y,z

Find all positive integers x, y, z satisfying $$x^{y^{z}} \cdot y^{z^{x}} \cdot z^{x^{y}}=5xyz$$ I took log on both sides, which led me to $$(\log x)(y^{z}-1)+(\log y)(z^{x}-1)+(\log z)(x^{y}-1)=\log ...
2
votes
1answer
77 views

Two questions on trigonommetric sums and integrals

Is it true that $\int_{0}^{\infty}\sin(mx)\sin(nx) \, dx = \delta (m-n) $ although using Euler formula I get a linear combination of $ \delta(m-n) $ and $ \delta (m+n)$? What is the sum ...
2
votes
1answer
39 views

Exponential function as a sum

I have an exercise that asks me to write $e^{2x}$ using a power series of $x+1$. I know that $$e^{2x}=\sum_{n=0}^{\infty}\frac{(2x)^{n}}{n!}$$ Then, I tried something like this $$x=y+1\Rightarrow ...
2
votes
1answer
58 views

Convergence of $\sum_{ k\in\mathbb{N} } \left( \frac{\lambda^k}{k!} \right)^n$

We know that $\sum_{ k\in\mathbb{N} } \frac{\lambda^k}{k!} = e^\lambda$. I'm interested in the convergence of $$S^{(n)}=\sum_{ k\in\mathbb{N} } \left( \frac{\lambda^k}{k!} \right)^n $$ for some value ...
2
votes
1answer
94 views

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, ...
2
votes
1answer
118 views

What is the infinite sum of $a^{b^x}$?

What would $$\sum^{\infty}_{n=0}(1/2)^{4^n}$$ be and how to determine it? EDIT: Apologies. I can see this converges by the ratio test. My issue is working out its sum, more for fun really. It ...
2
votes
4answers
44 views

Number of distinct real roots with $e^{-x}$ in the equation

How to find the number of distinct real roots of the equation $$13x^{13}-e^{-x}-1=0$$ I know that we generally find number of real roots by observing number of sign changes in $f(x)$ and $f(-x)$ but ...
2
votes
1answer
54 views

Median and Mean of Sum of Two Exponentials

I have a cumulative distribution function: $$G(x) = -ae^{-xb} - ce^{-xd}+h$$ The associated probability density function is: $$g(x) = abe^{-xb} + cde^{-xd}$$ My problem concerns $x\ge 0, X \in R$. I ...
2
votes
1answer
146 views

Integrating exponential of multiple exponentials

I have a integral term that looks similar to $\int_0^\infty\exp(-u-ae^{-c_1u}-be^{-c_2u})\,du$ where the constants $a,b,c_1,c_2>0$. For the case where $b=0$ I can use the answer from: Integrating ...
2
votes
0answers
80 views

Descartes Rule of Sign for exponential sums

I have the following exponential sums ($x\in\mathbb{R}$) $$f(x)=\sum_{i=1}^Na_iP_i(x)b_i^x$$ where $P(x)$ is some monomial, e.g., $x^2, x^3,\dots$, so $f(x)$ looks like ...
2
votes
0answers
78 views

Upper Bound on $\frac{1}{1-\beta u}-\sum\limits_{n=0}^{\infty}\frac{e^{-u}u^n}{n!(1-\beta(n+1))}$

Is there any procedure to find an upper bound of the following expression? $$\frac{1}{1-\beta u}-\sum_{n=0}^{\infty}\frac{e^{-u}u^n}{n!(1-\beta(n+1))}$$ Here $u,\beta\in\mathbb{R},\ u>1,\ ...
2
votes
1answer
91 views

The identity of a polynomial sum

I am wondering if there is a recursive formula to calculate $$S=1^{k}+2^{k}+3^{k}+\dots+n^{k}$$ Where $k$ and $n$ are natural numbers.
2
votes
0answers
43 views

a functional space for the study of exponential sum

I would like to study the exponential sum in an appropriate functional space. In particular: $f(x): \mathbb{R} \to \mathbb{R} $ $f(x) = \chi_K \sum_{i=1}^{M} R_i \exp{(s_i x)}$ where $R_i, s_i \in ...
2
votes
3answers
122 views

Need to check simplification of expression with infinite sum of exponentials

In reviewing a paper, I've come across a simplification the looks fishy to me, but I'm having a hard time checking it. I pulled out my old CRC handbook, but neither that nor Google are proving to be ...
1
vote
2answers
201 views

How many times do you need to double previous result to get at least $10^{82}$?

This is pretty straightforward, but I'd like to study, how find out, how many times do you need to double previous result of calculation to get some sum, for example: $10^{82}$ $1\times 2 = 2$ ...
1
vote
3answers
387 views

Summing a exponential series

What is the appropriate way to simplify such an expression. i am unsure of how to use the series i know to apply to this situation $$\sum_{L=0}^{M}s^{L}L^{2}$$ do i modify such a series as power ...
1
vote
2answers
48 views

What is the distribution of the average of IID exponential RVs? (have to use MGFs)

Just a quick question as I cannot figure this out. Here is the problem: Q: What is the distribution of the following? ($Y_1 + Y_2 + Y_3 +\cdots+ Y_n$) $ /n$ if each $Y$ is independent and identically ...
1
vote
2answers
47 views

Summation simplification on exponentiaion

I would like to ask help for the following question: Simplify $$\sum_{x=0}^{+\infty}{e^{-x/\mu}}$$ Will integration work here? Meaning, can I do it this way? ...
1
vote
2answers
83 views

How to solve these series?

Can anyone help me understand how to solve these two series? More than the solution I'm interested in understanding which process I should follow. Series 1: $$ \sum_{i = 3}^{\infty} i * a^{i-1}, ...
1
vote
1answer
129 views

Summation of roots of unity equal to zero

Let $u$ be a integer which has an odd prime $m$ as a divisor. So we let $u=u_1m$ where $u_1=\frac{u}{m}$. Consider the set of $u$th roots of unity, i.e., $A=\{e^{j\frac{2\pi}{u}n} | 0 \leq n \leq ...
1
vote
2answers
202 views

Reducing infinite summation

I'm trying to reduce the following nested summation (removing all summations), using the fact that: $\sum_{i=0}^\infty\ a^i = 1/(1- a)$. Problem: $\sum_{n=0}^\infty\sum_{m=n}^\infty\ a^nb^m$ I know ...
1
vote
1answer
445 views

Use Cauchy's Multiplication Theorem and the Binomial Theorem to prove $\exp(x+y)=\exp(x)\exp(y)$

I am to use Cauchy's Multiplication Theorem and the Binomial Theorem in order to prove $\exp(x+y)=\exp(x)\exp(y) $ but I have no idea where to begin. All I can think of doing is setting $\exp(x)$ ...
1
vote
1answer
338 views

Non trivial upper bound for an exponential sum

Suppose $h \in \mathbb{N}$, is there a known non trivial upper bound for $$\left| \frac{1}{n} \sum_{m=1}^n e^{2 \pi i h (2 \pi m)} \right|?$$
1
vote
3answers
88 views

Solve for Exponent in a summation

For a simple summation: $$ Z = \sum_{x=1}^X x^n $$ Solve for n. I have googled unsuccessfully and I do not remember this.
1
vote
1answer
58 views

The limit of a sum with elements from 2 sequences

Let's consider the sequence of real numbers $(a_{n}),n\geq1,a_{1}>0$ that satisfies the following recurrence: $$\frac{n(n+2)}{(n+1)}a_{n}-\frac{n^2-1}{n}a_{n+1}=n(n+1)a_{n}a_{n+1}$$ I'm supposed ...
1
vote
2answers
39 views

The result of exponential sum formula

I am awakard to deal with math problem I am trying to understand the first condition that is (k-r)=mN I can understand when (k-r) is mN, the left formula is 1 But I don't know how to left ...
1
vote
1answer
51 views

Absolute value of a cubic Gauss sum (over Field $\mathbb{F}_p$ )

I'm interested in the quantity $|\sum_{x \in \mathbb{F}_p} \omega^{ax^3+bx^2+cx}|$ where $a \in \mathbb{F}_p^*,b,c \in \mathbb{F}_p$ and $\omega$ is a primitive $p$-th root of unity i.e. ...
1
vote
4answers
76 views

closed-form term for this sum:

related to this question: Is there an easy closed-form term for $$\sum_{j=k}^{\infty} \frac{x^j}{j!}e^{-x},$$ thus when the sum starts at a constant $k$ instead of $1$? EDIT: Thanks for your help. ...
1
vote
1answer
52 views

Complex exponential sumation: $\sum_{-M}^M e^{-2\pi ikf}$

Does someone know how to prove that $\displaystyle {\sum_{k = -M}^M e^{-2\pi ikf} = \frac{sin((2M + 1)\pi f)}{sin(2\pi f)}}$ where $f$ is a constant such that $-\frac{1}{2} < f < \frac{1}{2}$ ...
1
vote
1answer
87 views

What is the $\sum\limits_{i=0}^{\ (\log_2(n))-1)}\frac{n}{2^i}$?

What is the value of the following sum: $$\sum\limits_{i=0}^{\ (\log_2(n))-1)}\frac{n}{2^i}$$ Can you show how to go about arriving at the answer?
1
vote
0answers
12 views

Evaluation of infinite sum related to problems in elasticity

I'm working on some problems with relation to elasticity (plate mechanics in specific) and while I've made some progress the following sum is giving me a hard time ...
1
vote
0answers
16 views

Help in deriving proof with exponential and little-oh

I'm working through a proof of the Kelly Criterion found at http://www.elem.com/~btilly/kelly-criterion/. In checking up on some of the derivations I can't follow the last step in this sequence ...
1
vote
0answers
39 views

please evaluate this sum

$$\sum^\infty_{n=1}\frac{(-1)^n}{n!}\times\frac{3^{a+n}}{a+n}-\sum^\infty_{n=1}\frac{(-1)^n}{n!}\times\frac{1}{a+n}$$ I need help evaluating the above, step by step would help. Thanks for a>0 or ...
1
vote
1answer
40 views

Is it possible to solve this nonlinear equation analytically?

Is it possible to solve the following equation analytically? $B_1\exp(\beta_1 x) + B_2\exp(\beta_2 x) = C_1\exp(\alpha_1 x) + C_2\exp(\alpha_2 x)$ where, $B_1$, $B_2$, $C_1$, $C_2$, $\beta_1$, ...