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Questions tagged [exponential-sum]

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

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Can anyone help me calculate this sum

$$\sum_1^{n}a^{p^{n-1}}$$ I tried many things but couldn't find it. I think I'm missing something and that it might be related to the multinomial theorem.
Souman's user avatar
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5 votes
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Generic bound on quadratic character sum

Let $\chi$ be the non-trivial quadratic character of $\mathbb{F}_q$, and let $f(x)$ be a square-free polynomial over $\mathbb{F}_q[x]$. Then by the Weil bound, we have the generic estimate $|\sum_{x\...
Madarb's user avatar
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2 votes
1 answer
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Enquiry on a claim in Titchmarsh. [closed]

There is a claim on p.107 of Titchmarsh's ''The theory of the Riemann zeta function'', that if $0<a<b \leq 2a$ and $t>0$, then the bound $$\sum_{a <n <b} n^{-\frac{1}{2}-it} \ll (a/t)^{...
MetricSpace's user avatar
0 votes
2 answers
89 views

A threshold for an exponential sum

I came across a sum where I have to find the smallest $n$ so that $$\sum_{x = 0}^n \frac{250^x}{x!} \ge \frac{e^{250}}{2}$$ I wrote a Java code and the result was 55 but with Desmos it was over 129 (...
Issaouik Aziz's user avatar
2 votes
1 answer
44 views

For the binomial Hopf algebra, what is the group of grouplike elements of the dual algebra?

The linear space of finite polynomials over a field $\mathbb{K}$ of zero characteristic has a structure of a graded connected bialgebra (meaning the zeroth subspace is isomorphic to the base field): $\...
Daigaku no Baku's user avatar
0 votes
1 answer
25 views

Order Statistics from a sum of exponential distributions

Let $X_i$ $(X_1, \dots, X_n)$ and $Y_i (Y_1, \dots,Y_n)$ be i.i.d. exponential r.vs with rate $\lambda$. Let $Z_i= X_i+Y_i$. How to write the pdf of the k-th order statistics of the $Z_i$ random ...
user9467051's user avatar
1 vote
1 answer
72 views

Double summ with exponential term

How to find an exact value of the following series $$\sum_{n=2}^{\infty}\frac{1}{2^n}\sum_{k=1}^{n-1}\frac{1}{k(n-k)}$$ I tried to reduce the problem to computing double integral. I also tried some ...
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37 views

How can I express two positive integers as the real and imaginary part of an exponential sum

In the following problem, suppose you have two positive integers $A$, $B$, $A$ odd, $B$ even and let $A^2+B^2=p$ a prime. Let $g$ be a primitive root modulo $n$. For any $b$ in $\mathbb{Z}^{\times}_n=\...
3809525720's user avatar
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118 views

$\prod_{n=1}^{\infty}(1+e^{-n})=^{?}$ and a series .

Conjecture : $$\prod_{n=1}^{\infty}(1+e^{-n})=^{?}1/1!+1/2!+1/3!+0/4!+1/5!+1/6!+7/7!+5/8!+9/9!+7/10!+\cdots+a_n/n!+\cdots$$ Where $a_n$ is an integer such that : $$0\leq a_n\leq n$$ Some arguments : ...
Miss and Mister cassoulet char's user avatar
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1 answer
98 views

Evaluating the infinite sum of $n^k/n!$ [duplicate]

I am trying to find a general formula for the following summation; $$S_k=\sum_{n=0}^∞ \dfrac{n^k}{n!}$$ So naturally, first I tried to find the expression for $k=1,2,3,4$ to see if I can get a pattern ...
Cognoscenti's user avatar
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41 views

Why softmax is preserving pattern when applied along different axes in a matrix?

I am calculating the softmax function over a matrix containing random float values using the following methods: row-wise column-wise Considering the whole matrix After calculating the values, I have ...
Nimantha's user avatar
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4 votes
2 answers
346 views

Exploring the convergence properties of a cost function involving orthogonal projection of one-hot vectors

$\newcommand{bm}[1]{\mathbf{#1}}$Given the semi-orthogonal fat matrix ${\bm B} \in\mathbb R^{c \times d}$ (i.e., $c\leq d$, $\bm {BB}^\top=\bm I$), the matrix $\bm X \in {\Bbb R}^{m \times n}$, $c$ ...
Phoenix's user avatar
  • 103
0 votes
2 answers
87 views

$\mid 1 + w + w \mid = \mid 1 + w^2 + w^2 \mid$ where $w=e_p(1)$

I was trying to digest something related to exponential sum however there was obstacle for me. My question is the following: $\mid \sum _{i=1} ^3 e_p(a_i) \mid$ where $e_p (a_i)=e^{\frac{2\pi.i}{p}a_i}...
Fuat Ray's user avatar
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3 votes
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If $\inf\{b\in\mathbb{R}\mid\sum_{n=1}^{\infty}e^{-ax_n-by_n}<+\infty\}=1-a$ for all $a\in [0,1]$, does this equality hold for all $a\in\mathbb{R}$?

Let $\left\{x_n\right\}_{n=1}^{+\infty},\left\{y_n\right\}_{n=1}^{+\infty}\subset [0,+\infty)$ be two sequences of non-negative real numbers. Suppose there exist $\lambda\ge 1, c\ge 0$ such that $\...
YC Su's user avatar
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1 vote
3 answers
85 views

Show that $\mid \sum _{i=1} ^n e^{\frac{2\pi.i}{p}a_i} \mid \ge n. \cos(\frac{2\pi}{p})$

I am currently struggling with exponential sum for finite fields and here is the question. Let $p$ be prime and $a_1, \dots,a_n \in \mathbb F_p$ such that $a_i \in [-1,1]$. Show that $\mid \sum _{i=1}...
Fuat Ray's user avatar
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0 votes
0 answers
23 views

Help solving Exponential Sum problem using KKT conditions

I am trying to solve the following convex optimization problem where $a_i, b_i >0$ for $i=1,2,3$. I am wondering if it is possible to get a general formula for the optimal solution without checking ...
Zona's user avatar
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0 answers
35 views

$ 0 = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{\ln(n)} \implies Re(s) \leq \frac{1}{2}$?

Define $f(s)$ as $$ f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{\ln(n)}$$ where we take the upper complex plane as everywhere analytic. Notice this is an antiderivative of the Riemann Zeta function, ...
mick's user avatar
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Quotient curve of Fermat curve over a finite field (Ireland -Rosen exercise)

I'm thinking a exercise 12 of Ireland-Rosen "A Classical Introduction to Modern Number Theory" p.226 as follows: But I have no idea, could you give some hints starting from (a) ?
user682141's user avatar
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2 votes
1 answer
55 views

Average of Sums of Exponentials

Suppose you have a function of the form $f(x)=\sum_{\mathcal{A}}e(nx)$, where $\mathcal{A}$ is some subset of the naturals and $e(x)=e^{2\pi ix}$. Now let $b$ and $q$ be coprime integers. I think that ...
Itachi's user avatar
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1 answer
80 views

How to find the sum of a finite number of increasing fractional powers

It is easy to show $\quad\sum_{i=1}^{\infty} \dfrac{1}{n^i}=\dfrac{1}{n-1}\quad$ e.g. $$\quad\sum_{i=1}^{\infty} \dfrac{1}{2^i}=\dfrac{1}{1}\quad$ $\quad\sum_{i=1}^{\infty} \dfrac{1}{3^i}=\dfrac{1}{2}...
poetasis's user avatar
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0 votes
1 answer
37 views

Zero set of double trigonometric polynomials.

Let $F\subseteq \mathbf{Z}^2$, ($\mathbf{Z}^2$ is the integer lattice), such that $|F| = \alpha < \infty$. For any $(m_1,m_2)\in \mathbf{Z}^2\setminus F$, we can define a double trigonometric ...
Doofenshmert's user avatar
5 votes
1 answer
127 views

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 ...
Habeeb M's user avatar
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Why are there multiple spirals in the Weyl-type sum?

This question comes from the Mathematica documentation. In the documentation of $e$ 1, it shows a neat example of Weyl-type sum: Letting $$ f(n;r,p) \triangleq \sum_{k=1}^{n} e^{i~r~k^p}, $$ then the ...
SY Z's user avatar
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1 vote
0 answers
26 views

Exponential sums related to abelian subgroups

I have a question here that might be very difficult, but perhaps someone who knows about groups and/or exponential sums might be able to help with it. Let $\Gamma$ be a finite group and $G_p\subset \...
JP McCarthy's user avatar
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48 views

About $T(n,k)=\frac{1}{(n+1)!}+\frac{1}{(n+2)!}+\frac{1}{(n+3)!}+...+\frac{1}{(n+k)!}$ where $n,k\in \mathbb{N}$

For $n,k\in \mathbb{N}$ let $$T(n,k)=\frac{1}{(n+1)!}+\frac{1}{(n+2)!}+\frac{1}{(n+3)!}+...+\frac{1}{(n+k)!}$$ Which of the below statement(s) hold true (A)$T(n,k)\geq \frac{1}{2}$ (B)$T(n,k)> \...
Maverick's user avatar
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5 votes
0 answers
107 views

Closed form for $\sum_{n=1}^\infty x^{-\frac{2\pi}{n}} e^{-2\pi n}$

I need a closed form for $$ \sum_{n=1}^\infty x^{-\frac{2\pi}{n}} e^{-2\pi n}$$ where $x\in[1,\infty)$ For $x=1$ we have the sum as $$ \sum_{n=1}^\infty e^{-2\pi n}=\frac{1}{e^{2\pi}-1}$$ For $1<x&...
Max's user avatar
  • 734
1 vote
1 answer
87 views

Sum of powers of the sum of all the square numbers in a factorial

Here's the exact question: The sum of all positive integers $p$ such that $13! \div p$ is a perfect square, can be written as $2^a . 3^b . 5^c . 7^d . 11^e . 13^f$ where $a, b, c, d, e$ and $f$ are ...
some random person's user avatar
3 votes
1 answer
66 views

Special property of the normal distribution density concerning infinite sums?

It seems that equi-distant Gaussian bell curves almost add up to the constant function $1$ already for relatively small sigmas, i.e. $$ \lim_{s\rightarrow\infty} \sum_{n = -\infty}^{\infty} \frac{1}{\...
coproc's user avatar
  • 1,608
1 vote
1 answer
76 views

Proof that $\sum_{y=0}^{\infty} \frac{x^y y}{y!} = xe^{x}$

I am looking for an analytical proof that : $$\sum_{y=0}^{\infty} \frac{x^y y}{y!} = xe^{x}$$ Both CAS Wolfram alpha and sympy agree on the result : ...
mocquin's user avatar
  • 253
1 vote
0 answers
55 views

How to estimate the exponential sum $\sum_{|m|\le N} \frac{e^{im^{3/2}}}{m^{1/4}}$

I am interested in upper-bounding tightly the following sum $$\mathcal{S}(N)=\sum_{|m|\le N} \frac{e^{im^{3/2}}}{m^{1/4}}$$ better than the naive triangle inequality estimate $$\mathcal{S}(N)\lesssim ...
Diffusion's user avatar
  • 5,601
0 votes
0 answers
28 views

Is there a general rule for addition or subtration of two numbers with the same base but different exponents?

I see the product and quotient rules in a quick online search, but I can't find amything for addition and subtraction. For example, a special case is: $$2^6 - 2^5 = 2^5$$ $$2^5 - 2^4 = 2^4$$ $$3^6 - 3^...
Andrew Singley's user avatar
3 votes
1 answer
219 views

Analytical evaluation of infinite series

I am trying to calculate the infinite series $$\sum _{n=-\infty }^{\infty } \frac{(-1)^{n+1} e^{-(n-1)^2\pi}}{1-e^{ (2 n-1)\pi}}\simeq -0.0903244354808$$ Are there any any analytical methods to ...
El Rafu's user avatar
  • 608
0 votes
0 answers
31 views

Exponential function to the power of a fraction

I have an equation such that $$1=e^{\frac{ylog(3+x)}{3+x}}-3 -x$$ Where e is the exponential function. How do I solve for x? According to the laws of exponents, $a^{m/n}=\sqrt[n]{a^m}$, Applying this ...
CorporateNationalism's user avatar
0 votes
2 answers
64 views

How to efficiently compute $\sum_{i=1}^n (x_i) ^ a$ for many different values of a and large n?

I need to compute $\sum_{i=1}^n (x_i) ^ a$ for many different values of a where n is large and x is fixed. I am in an environment where computing exponentials $(x_i) ^ a$ is effectively constant time. ...
Lalaland's user avatar
  • 175
0 votes
0 answers
70 views

Exponential sums reference

Looking for reference on exponential sums, in particular Jacobi, Gauss, Kloosterman and Ramanujan sums. The books mentioned in https://mathoverflow.net/questions/65429/exponential-sums-for-beginner ...
Ximenez's user avatar
  • 107
3 votes
1 answer
135 views

An infinite sum of products

I have to calculate this sum in closed form $$ \sum_{n=1}^\infty \prod_{k=1}^n \frac{x^{k-1}}{1 - x^k} $$ where $x < 1$. Numerical evaluation shows that this converges. The product can be performed ...
golfer's user avatar
  • 86
1 vote
1 answer
47 views

Bounding exponential sums

There is classical inequality that seems to often appear: $$\sum_{i=1}^N \exp(2i\pi n \theta) \leq \|\theta\|^{-1}$$ where $\|\theta\|$ denotes the distance to the closest integer. I do understand ...
Amomentum's user avatar
  • 385
0 votes
1 answer
30 views

Double EMA into Single EMA

The exponential moving average (EMA) operator is defined as: $$y_t(x, \lambda) = (1-\lambda) \sum_{i=0}^\infty \lambda^i x_{t-i}$$ where $1-\lambda$ is the normalization factor, and the operator is a ...
joeP's user avatar
  • 101
4 votes
0 answers
81 views

Vanishing of a certain double sum of roots of unity

For $d \in 2\mathbb{N}$ and $0 \leq k,l \leq d-1$ the following double sum turned up in my computations: $$ f(k,l)=\sum_{x=0}^{d-1} \sum_{y=0}^{d-1} \varepsilon(x+y)\exp\left( \frac{2 \pi i\cdot(k x + ...
Bipolar Minds's user avatar
2 votes
1 answer
105 views

Deriving an upper bound for an exponential function

I am studying risk theory this semester and we are currently covering the concept of adjustment coefficients, $R$. Essentially, the result we derived in class was that, for a random claim of size $x$, ...
Ethan Mark's user avatar
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0 votes
1 answer
97 views

Most efficient way to compute $S(n, a, b) = \sum_{i=1}^n a^i i^b$?

Given $n$, $a$ and $b$, find most efficient way to compute $S(n, a, b) = \sum_{i=1}^n a^i i^b$ Most trivial way would be $O(n\cdot(n+b))$. If we use fast exponentiation then it would be $O(n\cdot(\log ...
ishandutta2007's user avatar
0 votes
1 answer
73 views

Numerical convergence of a sum

I want to study the convergence of the function given below. $$ \sum_{n = 1}^{\infty} e^{-\Gamma^2 n^2} $$ The $n$ are integers! Here, if we check numerically, the function converges based on $\Gamma^...
CfourPiO's user avatar
  • 109
1 vote
1 answer
43 views

How to prove the inequality $|e^{2 \pi i \alpha x} - e^{2 \pi i \alpha y}| \ll \|\alpha(x-y)\|$.

How to prove the inequality $|e^{2 \pi i \alpha x} - e^{2 \pi i \alpha y}| \ll \|\alpha(x-y)\|$? Here $\|\alpha x\|$ is the distance from $\alpha x$ to the nearest integer. My attempt was to rewrite ...
trynalearn's user avatar
  • 1,661
1 vote
1 answer
135 views

Asymptotics for an exponential sum

I am trying to find an equivalent of : $\displaystyle{S_n=\sum_{k=1}^n}e^{i\sqrt k}$ I tried to elucidate the asymptotic behaviour of the subsequence : $$S_{n^2-1}=\sum_{k=1}^{n^2-1}e^{i\sqrt k}=\sum_{...
Adren's user avatar
  • 7,602
2 votes
2 answers
134 views

Solution (recurrence relation) of non-linear DE using the method of power series

I have to solve this non-linear DE $y' -e^y -x^2 = 0 , y(0)=c$ using powerseries. $y(x) = \sum_{n=0}^\infty a_{n}x^n $ $y'(x) = \sum_{n=1}^\infty na_{n}x^{n-1} $ so we get $\sum_{n=1}^\infty na_{n}x^{...
DontWorry's user avatar
  • 131
2 votes
1 answer
116 views

Evaluating an exponential sum along an arithmetic progression

I am familiar with the identity $$\sum_{1 \leq n \leq q}e^{2 \pi i n^2 / q} = \left( \frac{1+ (-i)^q}{1-i} \right)\sqrt{q}$$ I am wondering if there is a similar evaluation for the sum $$S(r):= \sum_{...
Daniel Flores's user avatar
1 vote
0 answers
31 views

Proving that periodic vector-valued function satisfies $f(\mathbf x)= f(\mathbf x')$ iff $\mathbf x' = \mathbf x + k + \mathbf am$

In order to create a layer for a neural network that uniquely encodes relative relationships between a set of angles, I'm looking for a differentiable function with some specific properties; I'm ...
monkeypuzzle's user avatar
0 votes
2 answers
60 views

How to show $\displaystyle\sum_{j=1}^n \cos\left((k+l)\frac{\pi (j-1/2)}{n}\right) = 0$

This result seems trivial, how would I show $$\sum_{j=1}^n \cos\left((k+l)\frac{\pi (j-1/2)}{n}\right) = 0$$ Where $0 \le k,l \le n-1$ And $k \neq l \neq 0$. I tried to use the exponential definition ...
Governor's user avatar
  • 469
0 votes
1 answer
346 views

Limit of summation of $\sum_{i=0}^{\infty} e^{-ik}$ series

I was wondering whether anyone was able to give some insight to determining the limit of the following summation $$ \sum_{i=0}^{\infty} e^{-ik} $$ If anyone has any idea or is able to point me in the ...
Christopher Briggs's user avatar
0 votes
0 answers
110 views

Upper bound for sum of powers of 2 from 1 to $\log n$ where powers are multiplied by 2 each time?

Upper bound for sum of powers of 2 from 1 to $\log n$ where powers are multiplied by 2 each time? I'm trying to find the upper bound for the following series $$s(n) = 2^1 + 2^2 + 2^4 + 2^8 + 2^{16} + ...
exponential's user avatar

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