Questions tagged [asymptotics]
Asymptotic behavior of functions, asymptotic series and related topics
946
questions
2
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A sequence linked to irrationality
Let $0 < c < 1$ be a real number and $ x \in \mathbb{R}$. We define the sequence $(u_n)_{n \in \mathbb{N}}$ by :
$$u_0 = x$$
$$ \mathrm{If}, u_n \le c, \mathrm{then}, u_{n + 1} = u_n + (1 - c) $$...
0
votes
0
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35
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Estimate for the length of a partial orbit for a shift map for which its delta neighbourhood covers an interval
Consider $f:[0,2\pi) \to [0,2\pi )$ given by $f(x) = (x + 1) \bmod 2\pi$ for all $x\in [0,2\pi )$, i.e. a shift map on the unit circle with anti-clockwise shift of $1$.
Denote the sequence $\{ x_n \}$ ...
5
votes
1
answer
350
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Lindelöf hypotheses for derivatives of zeta
The Lindelöf hypothesis says that if we have:
$$\zeta(\sigma+iT)=\mathcal O(T^a)$$
Then if one considers $\sigma=1/2$ then $\inf a=0$. Further, from convexity and the functional equation this implies ...
2
votes
0
answers
162
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Asymptotics of $\vartheta(x+y)-\vartheta(x)$, where $\vartheta$ is the Chebyshev function, when $y\in[x^\alpha, x]$ for some $\alpha\in(0,1)$
Introduction
Consider $\vartheta$ to be the Chebyshev function, that is, $\vartheta(x)$ denotes, for $x\in\mathbb N$, the sum $\sum_{p\le x,\ p\text{ prime}} \ln p$.
I am interested in asymptotics for ...
0
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0
answers
69
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Asymptotic expansion of generalised hypergeometric functions at infinity
I am looking for some resources that are available in public domain that have detailed description of the asymptotic expansion of generalised hypergeometric function ${}_pF_q(z)$ at infinity, i.e., $|...
0
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0
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56
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asymptotic expansions for $C^{1+\epsilon}$operators
I want to understand the asymptotic expansion for $C^{1+\epsilon}$ operators.
More precisely, if a complex operator $T(z)=\sum_{n \ge 1}T_n z^n$ is defined on a closed unit disk, and it is $C^{1+\...
0
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0
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29
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Multivariate delta method gradient calculation with mixed moments
Here's a slightly simplified version of my problem (using fewer dimensions than what I'm actually solving).
Take $X \in \mathbb{R}^2$. We already know that $\sqrt{n}(X - \mu) \overset{d}{\to} N_2(0,\...
12
votes
2
answers
887
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Asymptotics of a strange oscillatory function
Consider the function $f:\mathbb{R}\to \mathbb{R}$ defined by $f(x)=\sum_{n\geq 1}\sin(x/n^2)$. It is easy to see that $f(x) = O(\sqrt{x})$ for large real $x$. Is it true that
$f(x)>0$ for $x>0$...
0
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0
answers
58
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Multivariate Jackson inequality for Chebyshev approximation
There is an approximation of a multivariate function by a Chebyshev polynomial of degree n. One needs to understand how the approximation error behaves depending on the degree of the polynomial or ...
3
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0
answers
92
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Error function of the second moment of the divisor function
It is easy to show that the second moment of the divisor function has asymptotics:
$$\sum_{n\leq x} d_0(n)^2 = xP(\log(x))+E_2(x)$$
Where $P$ is some polynomial and that:
$$E_2 = o(x)$$
Previously, ...
2
votes
0
answers
101
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Asymptotic Independence of random walks from increments?
Suppose we have two random walks $(S_n:n\geq 1)$ and $(T_n:n\geq 1)$ building from independent identically distributed increment vectors $\{(X_k,Y_k):k\geq 1\}$, i.e. $S_n=\sum_{k=1}^n X_k, T_n=\sum_{...
1
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0
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65
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The intersection of $ n $ cylinders in $ 3D$ space
I posted the question on here, but received no answer
I recently found out about the Steinmetz Solids, obtained as the intersection of two or three cylinders of equal radius at right angles. If we set ...
2
votes
0
answers
134
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Beyond Watson's lemma
Suppose $f:[0,1]\rightarrow \mathbb{C}$ is a smooth function, which I wish to approximate near $0$. Watson's Lemma implies that I can find a smooth function $a:[0,1]\rightarrow \mathbb{C}$ such that:
$...
0
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2
answers
126
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Asymptotic bound of a simple alternating binomial sum
I'm a rather inexperienced researcher, I've been stuck on a question for a while. I would like to find the largest $N = f(n)$ that satisfies the following inequality:
$$\sum_{j = 0} ^ n p^{n - j} (-1)...
6
votes
2
answers
636
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On the asymptotic behaviour of the series $\sum_{n=1}^{\infty} \left ( \frac{\zeta(ns)}{n}+\frac{s}{1-ns}\right )$ near $s=0$
I am interested in determining the behaviour of the the series/function
$$f(s)=\sum_{n=1}^{\infty} \left ( \frac{\zeta(ns)}{n}+\frac{s}{1-ns}\right )$$
near $s=0$. It is clear that $f(0)$ is undefined....
1
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0
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180
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Asymptotic behaviour of a sum involving Möbius function
(This is a cross-posted simplification of this question posted in MSE which did not have a complete answer.)
I am trying to get the asymptotic behaviour when $n$ grows to infinity of a partial sum of ...
0
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0
answers
97
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+50
Is it possible to consistently and naturally define this subset of Hardy field?
Consider Hardy field $H$, the field of germs of functions at positive infinity.
Can we define $H_I\subset H$, such that it would have the following properties:
$H_I$ is an integral domain.
For each ...
5
votes
2
answers
251
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Small parameter expansion of an integral
I am trying analyze an integral of the form
$$I(\varepsilon)=\int_0^\infty f(t,\varepsilon) \,dt$$
where $\varepsilon$ is a small real parameter. The function $f(t,\varepsilon)$ is very complicated, ...
5
votes
1
answer
360
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Asymptotic solution of a system of ODEs
I have asked this question on math.stackexchange, however, have not got any answer. Therefore, I suspect that this system of ordinary differential equations cannot be solved analytically. But I still ...
4
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0
answers
163
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Growth rate of a recursively defined sequence
For a side project with a friend (having to do with fractional iterates of the exponential function), we're looking at a sequence $a_n$ defined recursively by equations $a_1 = 1$ and
$$a_n = \sum_{k=1}...
7
votes
1
answer
456
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On a paper by Dimitrie Pompéiu and on one (in two parts) by Edmund Landau
To celebrate the new year and the future of mathematics (or the mathematics of future), I see no better way to ask a question stemming from my researches on power series.
The two papers the title ...
0
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0
answers
20
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Asymptotic behavior of a truncated MGF of a quadratic form of a standard gaussian
Assume $X$ follows an $n$ dimensional standard Gaussian distribution and $A$ is an $n\times n$ semi-definite matrix. I am trying to analyze the asymptotic behavior of $\mathbb{E} e^{-X^TAX} \mathbf{1}...
2
votes
1
answer
90
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Show $v(x,t) \in L^2([0,T];H^2(\mathbb{R}))$ when $v(x,t)$ is a transformation of a $L^2([0,T];H^2(\mathbb{R}))$ function
Context: I am reading a paper on Long-Time Asymptotics of the thin film equations, in which the authors consider the strong solutions of the thin film equation in 1-D and transform them using a time-...
0
votes
1
answer
126
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Asymptotic behavior of the polylogarithm function and generalisation
So, right now I am writing my master thesis and I need to find a reference for a formula I found in a paper:
$$
\sum_{k=1}^{\infty}k^{-\alpha}(1-\varepsilon)^k\sim b+c\Gamma(1-\alpha)\varepsilon^{\...
0
votes
0
answers
43
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Asymptotic counts for imaginary quadratic discriminants with fixed splitting conditions
Let $p$ be prime and $r$ be a positive integer. I am interested in asymptotics for the number of imaginary quadratic discriminants $d$ such that
$p$ does not divide the conductor of $d$,
$p$ splits ...
10
votes
1
answer
314
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Does the mean ratio of the largest prime factor in prime gaps to the lower bound of the gap converge?
Posting in MO since this questions has been unanswered in MSE for 3 months.
Let $p_n$ be the $n$-th prime and $q_n$ be largest among all the prime factors of the composite numbers between $p_n$ and $...
2
votes
1
answer
191
views
Series with the smallest number whose square is divisible by $n$
I was looking into this sequence. And I'm particularly interested in the asymptotic behavior of the following series (which is stated on the site) $$\sum_{k=1}^n \frac{1}{a(k)} \sim \frac{3(\log n)^2}...
1
vote
0
answers
105
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On the derivation of some asymptotic expressions involving combinatorics
My questions come from the supplementary material in a recent preprint Nonequilibrium statistical mechanics of money/energy exchange models. My first question comes from page 35. Specifically, suppose ...
5
votes
1
answer
131
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History of asymptotic expansion of Laplace’s method between Laplace and Erdélyi
In 1774 Laplace understood that $I≔∫\textrm{d}x \exp kf(x)$ for $k≫0$ can be estimated if one knows 2-jet of $f$ at its point of maximum (as $I₀ ≔ ∫\textrm{d}x \exp kf₀(x)$ with $f₀$ quadratic with ...
0
votes
0
answers
174
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Why is the sign of the integration negative?
Let
\begin{aligned}
I=\int_0^1 B^{\frac{1}{1-\alpha + \alpha x}} x^{k - 1} \left(\frac{\alpha \log{\left(B \right)}}{(\alpha-k-1)^2} +\frac{1}{k} + \log{\left(x \right)} \right) dx,
\end{aligned}
...
1
vote
0
answers
99
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Asymptotic distribution of L infinity norm of Gaussian random vector
Let $\mathbf{X}_n = (X_{n,1}, \ldots, X_{n,n})$ be a $n$-dimensional random vector with $N_n( \mathbf{0}_n, \boldsymbol{\Sigma}_n )$ distribution. The asymptotic distribution of the $L_\infty$-norm of ...
22
votes
1
answer
4k
views
A challenging (for me) limit calculation
How to calculate the following limit
$$
\lim_{n\to\infty}\sqrt{n}\underbrace{{}\sin(\sin(\sin(\sin(\cdots\sin(\frac{1}{\sqrt{n}})\cdots))))}_{n \text{ sin's}} \text{?}
$$
${}{}$
0
votes
0
answers
103
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Existence of Green functions and some properties
Let $\Omega$ be a smooth domain in $\mathbb{R}^N$, $N\geq 3$, $p\in \Omega$ is a fixed point, $\lambda$ is a parameter (can be 0,>0,<0), if there exisits a Green function $G_{\lambda}(x,p)$ ...
5
votes
1
answer
447
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Is there an upper bound on the number of representations as a sum of squares?
I am interested in finding upper bounds for the Sum of squares function defined as $r_k(n) = |\{(a_1, a_2, \ldots, a_k) \in \mathbb{Z}^k \ : \ n = a_1^2 + a_2^2 + \cdots + a_k^2\}|$ whenever the ...
2
votes
1
answer
176
views
Sum of Bessel function with integer parameters and fixed argument
Question.
Let $J_{\nu}$ be a standart Bessel function of the first order. What is the asymptotic of the sum $\sum_{n\ge 0}|J_n(t)|$ as $t\to\infty$? An upper bounds stronger than $O(t)$ are also ...
0
votes
0
answers
102
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What is known about the average growth rate of the denominators of $n$ Egyptian fractions summing to one?
Motivation
In the following question posted here on MO and over at MSE, user Noah Schweber asks about a weighted count on Egyptian fraction representations (EFRs). To that end, he defines the ...
1
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0
answers
89
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Help understanding this proof of asymptotic bounds on solutions
In Boukanjime et al. " https://www.sciencedirect.com/science/article/pii/S0005109821004039 " I'm having difficulties understanding the proof of Theorem 5.1 after equation (13). It's a ...
4
votes
2
answers
408
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Binomial coefficient asymptotics
What is the probability that the number of heads in $n$ fair coin tosses is exactly $\lfloor n/2 + c\sqrt{n} \rfloor$
for $c \leq O(1)$, $n > \omega(1)$?
Of course the answer is
$$ \frac{1}{2^n} \...
3
votes
0
answers
118
views
How to find the right path of integration to get the asymptotic partition formula
I am trying to understand how the asymptotic partition formula $p(n) \sim \frac{e^{\pi\sqrt{\frac{2n}{3}}}}{4n\sqrt3} $ was derived for a project and have been reading and following many papers.
I am ...
1
vote
1
answer
104
views
$L^1$ error between indicator function and smoothed out version
For a large parameter $r>0$, consider the indicator function $1_{[-r,r]}$ and its convolution with the (normalized) Gaussian $\frac{1}{\sqrt{\pi}}e^{-x^2}$, that is,
$$f_r(x) = \frac{1}{\sqrt{\pi}}\...
2
votes
1
answer
183
views
Extreme case bounds on Diophantine approximation
I am wondering about the possible best case approximation and worst case approximation of irrational numbers. I think the appropriate formulation is whether there are functions $\hat{b}(q)$ and $\...
3
votes
0
answers
78
views
Diameters of permutation groups with transitive generators
Suppose that for a permutation puzzle on $n$ elements we have a subroutine to place any $m>3$ elements in arbitrary positions (possibly scrambling the rest). Can we solve the puzzle (if it is ...
1
vote
1
answer
79
views
Asymptotic scaling of mean and variance for the norm of random vector (non gaussian components)
The norm of a vector whoose components $X_i$ are normally distributed follows the Non-central chi distribution and it can be shown that, increasing the number of components $k$ (i.e. the dimension of ...
3
votes
0
answers
154
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Asymptotic distribution of the extreme, standardized order statistics of uniform distribution?
Let $\{U_{k, n}\}_{k=1}^n$, denote the order statistics of a sample of $n$ iid uniform $[0, 1]$ variates.
Note that, marginally $U_{k, n}$ is distributed $\mathrm{Beta}(k, n+1 -k)$.
Therefore, let us ...
2
votes
2
answers
209
views
Asymptotic scaling of mean and variance for non-central chi distribution
Define $Y \equiv \sqrt{\sum_{i=1}^k(\frac{ X_i}{\sigma_i})^2}$, with $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$ and independents.
It is known that $Y$ is distributed as a non-central chi (Noncentral ...
11
votes
1
answer
467
views
Asymptotics for pairs of positive integers whose harmonic (resp. geometric) mean is an integer
How does the cardinality of $$\{(a,b): 1 \leq a,b \leq n, \ 2ab/(a+b) \ \mbox{is an integer}\}$$ grow as a function of $n$? What about $$\{(a,b): 1 \leq a,b \leq n, \ \sqrt{ab} \ \mbox{is an integer}\}...
3
votes
1
answer
136
views
Asymptotic behavior of the integral that contains $\delta$ function
The integral I want to calculate is defined as
$$
P(s)=\int_{-\infty}^{\infty}{\rm d}x\int_{-\infty}^{\infty}{\rm d}y\ \delta\left(\frac{(x+y)^2+4x^2y^2}{(x+y)^2+(x+y)^4}-s\right)e^{-\left(x^2+y^2\...
2
votes
1
answer
692
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Does the Riemann hypothesis predict a bound for this prime-counting function?
Does the Riemann hypothesis predict an upper bound for
$$\left|f(x)-\left(\operatorname{li}(x)-\frac{x}{\log x} \right)\right|,\quad x\ge 2\tag{1}$$
where
$$f(x)=\sum\limits_{n=2}^x \frac{\Lambda(n)}{\...
2
votes
1
answer
257
views
$L^2$ norm of self-convolution
For $f\in [0,1]^n$ with $\sum_i f_i = 1$, define the self-convolution $g=f*f$ with
$$g_i = \sum_{j,k \mid j+k \equiv i \bmod n} f_j \cdot f_k$$
and define the L2-norm as
$$\|f\| = \sqrt{\sum_i f_i^2}.$...
3
votes
0
answers
225
views
Is this irregular curve asymptotic to $\log (2) (\log (x)-\log (2))^2-\frac{\log (2)}{2}$, or is the asymptotic something else?
Let $a(n)$ be the Dirichlet inverse of the Euler totient function:
$$a(n) = \sum\limits_{d|n} d \cdot \mu(d) \tag{1}$$
And let the matrix $T$ be:
$$T(n,k)=a(\gcd(n,k)) \tag{2}$$
which has the property ...