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

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1answer
23 views

Solving Recurrence Relations with generating functions when the variable is in the function.

I'm studying for a midterm and couldn't figure out these three recurrences that I came across: $i_{n+1}=2ni_n+2i_n+2$ with initial condition $i_0=1$ $j_{n+1}=3j_n+1$ with initial condition $j_1=1$ $...
-4
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0answers
67 views

Is the “Sophomore's dream” number rational or irrational?

I wanted to edit the wikipedia page of the "Sophomore's dream" page and wondered is the number rational, algebraic or transcendental (or belongs to another catagory) Link to wikipedia page https://en....
0
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1answer
43 views

Sum of a series with exponential and polynomial terms [closed]

I have reduced the expression that I am working on to the following sum of series, which is definitely converging. It would be great if someone can help me out with this or suggest ways this can be ...
0
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3answers
44 views

Does this sequence have a closed form representation?

We know that $$ \sum_{s=0}^\infty \frac{\lambda^{s}}{s!} = e^\lambda$$ Relatedly, $$ \sum_{s=1}^\infty \frac{\lambda^{s}}{s!}s = \lambda \sum_{s=1}^\infty \frac{\lambda^{s-1}}{(s-1)!}$$ For which ...
0
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0answers
14 views

Infinite exponential sum doubt

Hello! I have a couple of doubts regarding a formula seen here : $$\sum _{k=1}^{\infty } \frac {e^{kz}}{k}= -\log (1-e^{z}) /; Re(z)<0$$ What would happen if the real part of z Re(z) were equal ...
2
votes
1answer
58 views

Characters on rings of residue classes modulo polynomials over finite fields

First recall the following orthogonality relation on $\mathbb{Z}/n\mathbb{Z}$. Fix $n \in \mathbb{Z}$, $n \neq 0$. For $r \in \mathbb{Q}$, let $e(r) := e^{2 \pi i r}$. Let $x \in \mathbb{Z}$. Then ...
0
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1answer
51 views

Closed form of a series

I am looking for a closed form of the following convergent series: $$\sum_{n=0}^\infty \frac{(-\lambda^2)^n}{(6n+i)!}$$ For the case of $i=0$, the answer is ready, but when $i=1,2,3,4,5$, everything ...
2
votes
1answer
49 views

Generalised Gauss sums

Let $\chi$ be a non-trivial Dirichlet character modulo an odd prime $p$ and let $f(x) \in \mathbb{Z}[x]$ be a polynomial. We define the generalised Gauss sum $$ G(\chi, f):=\sum_{y \in \mathbb{F}_p^*} ...
1
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1answer
33 views

Linear exponential sum over $(\mathbb{Z}/(p^t))^*$

Let $p$ be a prime and $t$ a natural number. Let us denote $(\mathbb{Z}/(p^t))^*$ to be the group of units of $\mathbb{Z}/(p^t)$. I have the following exponential sum $$ S = \sum_{w \in (\mathbb{Z}/(...
0
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0answers
23 views

Calculation of an integral involving the sum of a range of natural exponential functions

Does somebody know how to solve the following integral, I extremely hope I can obtain its close-form solution: \begin{equation} \int \sqrt{ \sum_{i=1}^{M}\sum_{j=1}^{M} e^{-\frac{\frac{\left|\mathbf{...
0
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1answer
21 views

The upper bound of a sum of exponential function

Could someone help me to find the upper bound of the following function: $f(x) = \sqrt{\sum_{n=i}^{N} e^{-\alpha_{i}\cdot x}}$, where $x > 0$, the $i^{th}$ coefficient $\alpha_{i} > 0$. I got ...
2
votes
1answer
58 views

Solving an equation involving the trace of a field

Let $F$ be a finite field of order $q$ where $q=2^{n}$ and fix $l\in F\setminus{0}$ with $Tr(l)=0$. I want to determine the number of $a$ such that $$Tr(la)=Tr(la^{-1})=1,$$ where $Tr$ denotes the ...
0
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0answers
19 views

series summation of quadratic exponentials

I need to prove the equation $\sum_{n=-\infty}^{\infty} e^{-\frac{1}{2} An^2+iBn} = \sqrt{\frac{2\pi}{A}} \sum_{l=-\infty}^{\infty} e^{-\frac{1}{2A}(B-2\pi l)^2}$, where A and B are constants(possibly ...
0
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1answer
17 views

Calculating $\frac{e^{\frac{z_1}{T}}}{e^{\frac{z_1}{T}} + e^{\frac{z_2}{T}}}$ from $\frac{e^{z_1}}{e^{z_1} + e^{z_2}}$

Let's say I know the value of $\frac{e^{z_1}}{e^{z_1} + e^{z_2}} = x $ (say) Now I want to get the value of $\frac{e^{\frac{z_1}{T}}}{e^{\frac{z_1}{T}} + e^{\frac{z_2}{T}}} = y $ (say), where $T \...
1
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1answer
27 views

Cardinal exponentiation question of infinite cardinals.

I got a little confused with a question about cardinal exponentiation: Let $\beta$ be an ordinal, and let $K_\alpha$ , $\alpha < \beta$, be infinite cardinals with $K= \sum_{\alpha<\beta}K_\...
1
vote
1answer
22 views

Convergence of a convergence series with $e^{in}$

I am facing difficulties with this question: It says show by using the comparison test that the folowing complex series converges: $$\sum_{n=1}^\infty \frac{\Re(e^{in\phi})}{2^n}$$ The $\Re $ refers ...
3
votes
1answer
97 views

How to solve $6^{x} + 8^{x} + 9^{x} = 12^{x}$?

I checked on WolframAlpha and it says the answer is approx. 2.56639, but I don't know how to go about solving this equation. What kind of approach should I use?
1
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1answer
29 views

best values in the estimate of Vinogradov-Korobov

Let $C(N)=\sum_{1<n\le N}{n^{-it}}$. Vinogradov- Korobov estimate is $$|C(N)| \le KN\exp\left(-\gamma \frac{\ln^3 N}{\ln^2 t}\right).$$ What are the best values of K and $\gamma$ ? I have only ...
1
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1answer
29 views

How can we prove that the discrete Fourier transforms preserves the inner product up to a constant factor?

Let $d\in\mathbb N$, $\omega\in\mathbb C$ be a primitive $n$-th root of unity and $$\operatorname{DFT}_\omega:\mathbb C^d\to\mathbb C^d\;,\;\;\;z\mapsto\left(f_z\left(\omega^0\right),\ldots,f_z\left(\...
0
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1answer
55 views

Bound a complex exponential sum when we can only estimate its argument

Suppose we have a function $f : \mathbb{N_0} \to \mathbb{R}$ for which we can give an estimate of its values, and say its values $f(n)$ are roughly uniformly distributed for $n$ in some range $[1,N]$. ...
2
votes
1answer
187 views

upper bound formula the binomial coefficients with real valued arguments

I'm trying to prove the following. Let $n\in\mathbb{N},m\in\mathbb{N}\cup\{0\},\alpha\in (n-1,n)$ and $N\in\mathbb{N}:N\ge m+1$. Prove that \begin{align} &\sum_{k=N+1}^\infty\Big{|}\binom{n+m-\...
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votes
2answers
26 views

Most logical thing to do with these exponents and sums?

I'm doing homework for a programming class and came across this problem. There's no directions besides what I've shown, so I don't even know what it's asking me to do. What makes the most sense for ...
26
votes
1answer
452 views

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 $(...
2
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0answers
51 views

Subtle error with a module endormorphism on $\mathbb{Z}_8 \times \mathbb{Z}_8$

Let $a,b,c$ be arbitrary integers such that $a$ is odd and $(a,b,c)=1$. Let $R = \mathbb{Z}_8$, the set of all integer residues modulo $8$. Define an $R$-module endomorphism $\phi \colon R \times R$ ...
0
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4answers
53 views

If $A^{2x}=4$ what is ${A^{3x}-A^{-3x}\over A^x - A^{-x}}$

If $A^{2x}=4$ and $A > 0$, what is the numerical value of $${A^{3x}-A^{-3x}\over A^x - A^{-x}}$$ Could anyone find the solution and answer? Thanks
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0answers
26 views

When is the sum of three reduced rationals equal to an integer

When is the sum of three reduced rationals equal to an integer? This may be a duplicate of Question 1550437 but even if it is, there is no answer associated with this other question. Given three ...
1
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0answers
54 views

Matrix power eventually reaches partial order

Given an $n$ by $n$ matrix $A$ of non-negative integers and a column matrix $x$ of non-negative integers and of length $n$, and a partial order on the set $\{1, 2, 3, 4,\ldots, n \}$ (representing the ...
0
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1answer
37 views

Calculating Maximum Population Size After 106 Years, Starting With 3 Breeding Pairs

I have been looking all over for a formula or program to determine a maximum possible human population size within a $106$ year span. This specifically is in response the the claim by the Creationist ...
2
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0answers
52 views

Does this series converge? If so, to what?

i was solving some integral equations and some of them gave series whose convergence am not very sure of. Problem, if anyone can point how or to what the series converges, i will be more than glad. Am ...
0
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1answer
29 views

Notation for a specific summation

I'm trying to make the sum stop before the summation has a negative exponent. For example, I would want the sum to stop at $2^0$ in $2^3+2^2+2^1+2^0+2^{-1}$ The sum I'm dealing with is $$\sum P\...
0
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1answer
28 views

Generate exponential weights (sum of all = 1)

I have $500$ observations and I want to make exponential weighted average of them. I want the weights to be something like $w_i = 0.999^t$ when $t$ is from $1$ to $500$ (num of observations). ...
10
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2answers
594 views

Sum of factors of a huge number.

I recently appeared in a math olympiad and it had this one question which had me stumped. This was a few weeks back and I have been looking for a way to find its answer ever since, but with no success....
1
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2answers
94 views

How to simplify a sum of exponential equation?

Suppose I have three constants $a, b, c\in R$. I have a formulation as $f=e^{ab}+e^{ac}$. Can I have some result like $f'=e^{a(b+c)}$. I know $f'$ does not hold. But I just want to combine the two ...
1
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0answers
17 views

How to simplify a sum for the total cost of a yearly payment including compound interest

I want to simplify the below sum for the total cost over a yearly payment including compound interest over n years. An example: we have 150 euros that need to be paid every year and an interest of 2%....
1
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0answers
17 views

Relation between the sum of the values of a polynomial $f$ over a finite field, and the additive character sum with $f$ as the polynomial argument

Let $F$ be a finite field, let $f(T) \in F[T]$ and let $\psi$ be the canonical additive character of $F$. If $\sum_{x \in F}f(x) = 0$, what can we say of $\sum_{x \in F} \psi(f(x))$?
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0answers
26 views

Can every constant be written in that way?

When we are working in the ring $\mathbb{C}[e^{\lambda x} \mid \lambda \in \mathbb{C}]$ where the elements are of the form $$\sum_{i=0}^n \alpha_i e^{\lambda_i x}$$ where $\alpha_i, \lambda_i \in \...
1
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0answers
35 views

A Fact Stated in Davenport's Multiplicative Number Theory

In his text Multiplicative Number Theory on page 9, Davenport mentions that another means of expanding the L-function is known and then mentions the fact that, $$ \mathcal{F} \sum_{n=1,n \; odd} \...
0
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1answer
12 views

Summation of gaussians

Suppose we have given constants $A_i, x_i (i=1..N)$ Is it possible to approximately calculate the sum of N gaussians in less than N iterations for any x? (may be with some preprocessing) $$\sum_{i=1}...
0
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1answer
44 views

Solving for coefficients in half sum of two exponential functions

Let $$G(x) = \frac{\exp(-ax) + \exp(-bx)}{2}$$ I need to find the coefficients $a$ and $b$. Given: A table of 10 values of x and their corresponding G(x) values. I need some help in figuring out ...
0
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1answer
42 views

Prove theorem that concerns to sums of exponentials.

I try to solve this theorem: Theorem: Let $x_1, x_2,...$ iid with exponential distribution with rate $\lambda$. The density function of $S_n$ is given by: $$ P(S_n \le t)=\sum_{k=n}^{\infty}\frac{e^{-...
1
vote
3answers
101 views

Prove $\sum_{i=0}^n {{n \choose i} \times 2^i} = 3^n $ [duplicate]

I have to prove that $$\sum_{i=0}^n {{n \choose i} \times 2^i} = 3^n $$ Such that ${n \choose i} = \frac{n!}{i!(n-i)!} $ and $n $ is some arbitrary int I proved we can expand 2^I in a way such that $...
1
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2answers
79 views

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?
0
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0answers
50 views

Average of additive character sum over all polynomials of degree at most $d$ over $GF(q)$

Let $F$ be a finite field with $q$ elements, let $E$ be its degree $n \geq 2$ extension and let $\psi$ be the canonical additive character of $E$. For $d \leq n $ let $$ \mathcal{P}_{\leq d} = \{f(x) ...
0
votes
1answer
26 views

binary search tree size calculation

how do I calculate the number of elements a binary search tree can hold if have the height? For example a tree of height 3 can have 7 elements (7=1+2+4), and a tree of height 4 can have 15 elements (...
0
votes
0answers
21 views

Additive character sum over additive subgroups of finite fields, with special monomial arguments

Let $F$ be a finite field with $q = p^n$ elements, let $\psi$ be a non-trivial additive character of $F$, let $m$ be an integer coprime to $q-1$, and let $K$ be a large subspace of $F$, say $K$ is the ...
3
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0answers
45 views

Additive character sum over intersection of additive and multiplicative subgroups of finite fields

Let $H$ be a multiplicative subgroup of the finite field $\mathbb{F}_q$ with $q$ elements, say $H$ is the subgroup of $d$-th powers, $d \mid q-1$. Let $L$ be a subspace of $\mathbb{F}_q$ over some ...
2
votes
3answers
77 views

How to find the value of this expression?

I just saw this question in one exam. Please help me solve it. I am not able to find any clue on where to begin. (ignore that tick it might be wrong)
3
votes
1answer
47 views

A more general Kloosterman-type sum

Let $\mathbb{F}_q$ be a finite field and let $a,b \in \mathbb{F}_q$ not both zero. Let $\psi$ be the canonical additive character on $\mathbb{F}_q$. The classical Kloosterman sum is given by $$ K(a,b) ...
0
votes
1answer
59 views

How to find the solution to this summation

This was a question asked in our exam and we have to write a code for it. We have to find the summation of following series $\log(\sum_1^n (e^{x_i}))$ where $1 < n < 10^6$ and $0 < x_i < ...
3
votes
3answers
111 views

solve equation with sum $\ln x-\ln(\sqrt[2]{ x})+\ln(\sqrt[4]{ x})-\ln(\sqrt[8]{x})+…=2$

How to solve this? Any advice? $$\ln x-\ln(\sqrt[2]{ x})+\ln(\sqrt[4]{ x})-\ln(\sqrt[8]{x})+...=2$$ Next step I do this $\sum\limits_{n=0}^\mathbb{\infty}(-1)^n\ln(x^{\frac{1}{2^n}}) = 2$ But I don't ...