Questions tagged [exponential-sum]

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

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24
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0answers
501 views

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 ...
7
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1answer
75 views

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 $a,b,c\in\mathbb{N}_{>0},\,$ ...
8
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0answers
200 views
+250

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_{...
1
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1answer
39 views

Equidistribution of powers of primitive roots modulo $p$

Let me start with a nice experimental observation. Fix a large prime, say $p = 5003$. It turns out that $g = 2$ is a primitive root mod $p$. If we plot the powers of $g \in \Bbb F_p^{\times}$ (...
7
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1answer
89 views

Deciding convergence/divergence of $\sum_{m \geq 1} \frac{1}{m^3} \sum\limits_{\substack{k=1\\(m,k) = 1}}^m k \sin\left(\frac{2\pi k n}{m}\right)$

Let $n$ be a positive integer. I am attempting to determine whether the series $$ \sum_{m \geq 1} \frac{1}{m^3} \sum_{\substack{k=1\\(m,k) = 1}}^m k \sin\left(\frac{2\pi k n}{m}\right) $$ converges or ...
0
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1answer
38 views

Exponential Sum Approximation

Is it possible to show mathematically that, for a short portion, sum of two decaying exponential can be approximated by a single decaying exponential? i.e. $Ae^{-ax}+Be^{-bx}\approx Ce^{-cx}$?
0
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1answer
44 views

Sum of discrete Gaussian series

Is there a sum formula for the following sequence? It is a Gaussian with $N$ terms, $x_{0}$ and $\sigma$ are real numbers, with $0<x_{0}<N$ $$\sum_{x=0}^{N-1} e^{\frac{-1}{\sigma} \:(x-x_{0})^{...
11
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1answer
175 views

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 ...
3
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1answer
27 views

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 ...
0
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3answers
46 views

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 ...
1
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1answer
38 views

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 ...
1
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1answer
69 views

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 ...
0
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1answer
32 views

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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0answers
19 views

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 ...
0
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2answers
46 views

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 ...
0
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1answer
59 views

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 ...
3
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1answer
114 views

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 ...
0
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2answers
79 views

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 ...
0
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3answers
40 views

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 ...
1
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1answer
69 views

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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0answers
30 views

Do we know the Mobius inverse of e^x?

I'm refereing to the The Generalized Möbius Inversion: Suppose F(x) and G(x) are complex-valued functions defined on the interval [1,∞) such that $$ G(x)=\sum _{1\leq n\leq x}F\left({\frac {x}{n}}\...
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0answers
70 views

Does this sum $(1-\frac{1}{2^2})^{(1-\frac{1}{3^2})^{…^{(1-\frac{1}{p^2})}}}$ also related to Riemann zeta function?

I'm interesting for iterated exponention sum of the form $(z_1)^{z_2)^{...^{z_k}}}$ such that $z_1$ and $z_2$, $z_k$ are differents real exponents , This kind of sum was studied by many Authors such ...
1
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2answers
72 views

Absolute value of an exponential sum

Consider the following sequence $$ x_n=\Bigl|\sum_{t=1}^n\exp\Bigl\{2\pi i\Bigl[\frac1d-\frac {\lfloor n/d\rfloor}n\Bigr]t\Bigr\}\Bigr| $$ with some fixed positive integer $d$ such that $d<n$. I am ...
0
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1answer
32 views

Finding whole number solution to a multi variable nonlinear equation

$$\sum_{n=0}^z 10^n = 3^x*7^y$$ Given the equation above, how do I find solutions where $x,y,z$ are all whole numbers and $z\neq0$. Or if no such solutions exist how to prove it? In other words I'm ...
0
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0answers
33 views

Finding a sum of certain pth roots of unity

Say $p\equiv 1\pmod n$ and $a$ is a primitive root$\pmod p$. Can we find or estimate the value of $\sum_{i=1}^{(p-1)/n} \zeta^{a^i}$ and its conjugates? Note that if $n$ is odd, this sum is real by ...
2
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0answers
35 views

Find all complex $z$ such that $z=\sum_{n=2}^{\infty}\sum_{k=1}^{n}e^{\frac{2\pi ikz}{n}}$

Here is what I tried: The inner sum is a geometric series so $$\sum_{k=1}^{n}e^{\frac{2\pi ikz}{n}} =\frac{e^{c}\left(e^{nc}-1\right)}{e^{c}-1}$$ where $c=\frac{2\pi iz}{n}$ and $nc=2\pi i z$. So the ...
1
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1answer
47 views

Problem in the derivation of replica partition function of Hopfield network.

While I was following the derivation of the replica partition function of Hopfield network from here, I have found something strange explanation. Due to the independence of $s_{i}$'s, the following ...
0
votes
0answers
13 views

Upperbound for sum of two exponentials

Let $n \in \mathbb{N}$ and $x > 0, C > 1$. Consider the following $$2 \exp(-\frac{1}{2}(nx - \sqrt{2nx+1} + 1)) + \exp(-\frac{x^2n}{C})$$ How can I find an upper bound for the above sum in the ...
2
votes
1answer
69 views

Some identity related to Euler's identity

Let $n\geq 2$, and $z=e^{\frac{2\pi i}{n}}$. Then, for $1\leq m\leq n-1$ we have the identity: $$ \sum_{j=0}^{n-1}z^{jm}=0. $$ Considering a proof of the identity, we can use an idea about periodicity ...
0
votes
1answer
28 views

Is there a closed-form solution for this summation?

Context: I want to find the amount of time it takes for the sum of exponential growth to reach a given value. $x_t=\sum\limits_{i=0}^{t}x_0(1+r)^i$ In other words, is there a formula I can use to ...
2
votes
2answers
57 views

Sum of a series similar to exponential series [closed]

Is there a closed form expression or an $\textbf{approximation}$ for the below summation: $\sum_{x=0}^{\infty} \frac{\lambda^{kx}}{{(kx)!}}$ where k is a constant. I know that if $k=1$, ...
0
votes
2answers
56 views

Sum of Legendre sequence $S=\sum_{x=0}^{p-1}\left(\frac{x(x+k)}{p}\right)$

Calculate the sum of Legendre sequence $S(x):=\displaystyle \sum_{x=0}^{p-1}\left(\dfrac{x(x+k)}{p}\right)$ with $p > 3$ prime number, $k \in \mathbb{N}$ and $\text{gcd}(p,k)=1$. I have tried to ...
3
votes
1answer
74 views

Sum of series with exponential denominator

How do I calculate the sum $$ \sum_{j=1}^\infty\frac{j^3}{1+e^{2\pi j z}} $$ where $z\in\mathbb C$ such that $\mathrm{Re}\left(z\right)>0$. Can it be evaluated/expressed in terms of some special ...
0
votes
1answer
21 views

Explain on why the require time is not linear.

Here is the question: I got capital X, and I will compound interest (4%) to roll the capital, until the the interest grow > 3000. So, I start from 6000 (X), here ...
1
vote
1answer
52 views

Generalizing the formula for $\Lambda_k=\sum_{n=0}^{\infty} \frac{n^k}{n!}$

Let $\Lambda_k=\sum_{n=0}^{\infty} \frac{n^k}{n!}$. I was practicing evaluating exponential series, and I encountered the following few series $\Lambda_2=2e, \Lambda_3=5e$ and $\Lambda_4=15e$ which ...
1
vote
2answers
34 views

Geometric sum with complex exponent

I've encountered an infinite geometric sum while working a question: $$ (1 - e^{-6wi}) \sum_{0}^{\infty} {e^{-iwn}} $$ According to the answer sheet, this should resolve to: $$ \frac{(1 - e^{-6wi})}{(...
0
votes
0answers
32 views

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 ...
0
votes
0answers
25 views

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 : ...
1
vote
1answer
31 views

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 ...
0
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0answers
13 views

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 ...
0
votes
3answers
33 views

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 ...
-1
votes
1answer
24 views

Exponential formula with decreasing multiplication factor

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}^...
-1
votes
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 $4$, ...
-3
votes
1answer
86 views

Infinite sum of cos(ln(n))/n [closed]

What would be the value of the infinite sum $$\sum_{n=1}^\infty\frac{\cos\ln n}{n}$$
1
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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+...
4
votes
2answers
95 views

Finding a closed form solution for an infinite sum

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 ...
0
votes
0answers
45 views

Variance of the exponents and sum of exponentials

Let's assume we have two sets of non-negative real-valued numbers $\{a_i\}_{i=1}^{N}$ and $\{b_i\}_{i=1}^{N}$ ($N$ can be large) such that $\frac{1}{N}\sum_{i=1}^{N}a_i=\frac{1}{N}\sum_{i=1}^{N}b_i=...
6
votes
1answer
119 views

What is the series $\sum_{n=1}^{\infty} \frac{e^{-n^2 x}}{n}$?

Following Passare: How to compute $\sum 1/n^2$ by solving triangles I tried the following $$ \int_0^{\infty}\frac{e^{-nx}}{n^2} dx = \frac{1}{n^3} $$ So we can write (with some help of Wolfram Alpha) ...
0
votes
1answer
58 views

Inequality involving double summation

I have the inequality $$e^{-10}\sum_{a=0}^{n-i}a\sum_{b=0}^{n-1}e^{-abn(a-2bn)}<\ln n$$ A table in Mathematica quickly shows that the largest integer $n$ for which the inequality holds is $n=2$. ...
0
votes
1answer
48 views

Solve inequality involving double summation, exponentiation and $\ln$

I have the following inequality (where $n$ is a real number): $$\sum_{a=1}^n\sum_{b=1}^n\frac{a^n}{n^2}e^{-n-ba}\ge\ln n$$ Computation suggests this holds for $n$ greater than or equal to a number ...

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