# Questions tagged [exponential-sum]

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

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### A Nice Problem In Additive Number Theory

$\color{red}{\mathrm{Problem:}}$ $n\geq3$ 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 ...
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### 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 $a,b,c\in\mathbb{N}_{>0},\,$ ...
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### 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 ...
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### Absolute value of sum of additive characters of $\mathbb{F}_p$

Consider the absolute value of the following exponential sum: $\left|\sum_{x \in \mathbb{F}_p} \sum_{y \in \mathbb{F}_p}e^{\frac{2\pi i}{p}(ux+vy-wxy)}\right|$ for given $u,v,w\in\mathbb{F}_p$ with ...
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### What would the infinite sum of c^n/n converge to?

I know for example that the infinite sum of $c^n$ can be calculated when $|c|<1$ as below: \begin{equation*} \displaystyle \sum\limits_{n=0}^{\infty} c^n=\frac{1}{1-c} \end{equation*} And that ...
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### Formula or asymptotic behavior of a partial sum

I'm wondering if there is a known formula for the partial sum given by $$\sum_{k=1}^n e^{\sqrt{k}}$$ If not, could someone explain how one might deduce the asymptotic behavior of this sum? For ...
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### A special type of Gauss sum

In the work of my thesis I came up with a problem that is elementary, but I can't figure out its proof. Let $p$ be an odd prime, let $(\mathbb{Z}/p^n\mathbb{Z})^\times$ denote the multiplicative ...
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### How to write out this result? 10^116.75

I entered the following into my calculator: 10^116.75 And from a Google Search, I landed on this: 5.623413e+116
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### Evaluating infinite series of the form $\sum_{n=0}^{\infty}\dfrac{1}{P(n)}{\rm e}^{-Q(n)}$ where $P(n)$ and $Q(n)$ are polynomials

A problem has come up in my own work which involves the infinite series $$\sum_{n=0}^{\infty}\dfrac{1}{P(n)}{\rm e}^{-Q(n)}$$ where $P(n)$ and $Q(n$) are both polynomials in $n$. I am particularly ...
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### Is the sum of this series finite?

Let $a\in\mathbb{R}_{+}$. I consider the series $$f(q) = \left(\sum_{k=1}^q e^{-a(q-k)}\frac{1}{k}\right)^2$$ and the sum $$\sum_{q=1}^{\infty}f(q).$$ My numerical simulations show that this ...
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### How to calculate the exponential function

How to calculate the below equation $$\ln \frac{e^{x_1}+e^{x_2}+....+e^{x_n}}{n}$$ where $x_i, i=1...n$ and $n$ are some known values. In addition, $n=200$ and $x_i \in [1000, 2000]$. Or could we ...
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### Calculate the integral $\int_0^{2\pi}\sum_{k=n}^{\infty}\frac{e^{i(k-m)\theta}}{k+1}d\theta$

I would like to calculate this integral: $$\int_0^{2\pi}\sum_{k=n}^{\infty}\frac{e^{i(k-m)\theta}}{k+1}d\theta$$ where $n$ and $m$ belong to $\mathbb{N}$. My attempt: Instead of considering an ...
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### Is an exponential function strictly increasing?

Let $a, b$ and $c$ are nonnegative real numbers such that $a \geq b+c$, then I want to show that $a^r \geq b^r + c^r$ for all $r \geq 1$. For this I need to show that for $r \geq 1$, the ...
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### How to simplify √x when used as exponent?

Could anyone help me understand how to simplify the following expression. $$x^\sqrt x$$ If there was a number instead of $\sqrt x$ as an exponent, it wouldn't be a problem for me. But I have never ...
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### What does the $\ll$ operator signify? This paper says “order” but why didn't they use the big or little Oh notation? is the same thing?

https://www.cmi.ac.in/~shreejit/Zeta-Function.pdf "Our goal in this note is to discuss the behaviour of $\zeta(s)$ in the critical strip $0\leq\sigma\leq1$. More precisely, we want to inspect ...
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### Can I write an integral representation for this sum?

I need to compute the following sum, where $m=\frac nN$. $$\lim_{N \rightarrow \infty} \sum\limits_{n=1}^{N-1} \frac{1}{n^4(1+N^{-1}+m)^4} \exp \left\{ \left( 1+ m+ m^2 \right) \right\}$$ or even ...
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### Alternate representation for an exponential sum

Consider the following exponential series: $$c_q(n) = \sum_{1\leqslant k \leqslant q (k ; q)=1}e^{2πi(k/q)n}$$ Questions : (1) What are some alternate possible representation/s ( if possible : ...
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### Mobius function and exponential sums

It is easy to show that if $a$ and $n$ are positive integers with $\gcd(a,n)=1$, then $$\sum_{\substack{z=0 \\ \gcd(z,n)=1}}^{n-1} e^{2\pi i \frac{az}n} = \mu(n).$$ What is the general form of ...
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### Non-Assignment Question: Would like help finding cumulative # of hours spent in a 16-year period.

Hope you're doing well. PROPOSAL So, I’ve been trying to figure out how many hours of relief a clinic that I'm trying to help out has given to patients over a 16-year period. Each person treated by ...
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### How is the last step of this sum derived?

From the book Reinforcement Learning: An Introduction page 108 In the final sum I can see where the $0.1$ in front and the $0.9^k$ in the sum come from, but I can't see how the $2^k$ and 2 come ...
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I am attempting to simply the following calculation where the initial number is $2.4$ and add a decrease of $10$% from the previous step: $1.$ $2.4\times 0.9 = 2.16$ $2.$ $2.4\times\left(0.9 + {0.9}^... 1answer 46 views ### Writing an exponential formula where the factor decreases by$10%$each time I am attempting to write a formula for the following: if Input$0 - 50$Result is$1$, if Input$51 - 120$Result is$2$, if Input$121 - 271$Result is$3$, if Input$271 - 579$Result is$4$, ... 1answer 86 views ### Infinite sum of cos(ln(n))/n [closed] What would be the value of the infinite sum $$\sum_{n=1}^\infty\frac{\cos\ln n}{n}$$ 1answer 46 views ### Joint density of$Z:= (X_1 - a)_+ + (X_2 - a)_+…+ (X_n - a)_+$and$Y:=X_1+X_2+…+X_n$Given$X_1, X_2...X_n$are$i.i.d.$exponentially distributed random variables,$a$is an non-negative constant, find the joint density of$Z:= (X_1 - a)_+ + (X_2 - a)_+...+ (X_n - a)_+$and$Y:=X_1+...
I've come across a infinite series for which I've had difficulty finding a closed form solution: $$\sum_{i=1}^\infty \sin^2(\pi/i).$$ I believe that the series does converge and I've tried looking ...