Questions tagged [asymptotics]

Asymptotic behavior of functions, asymptotic series and related topics

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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 : $$u_0 = x$$ $$ \mathrm{If}, u_n \le c, \mathrm{then}, u_{n + 1} = u_n + (1 - c) $$...
Azoth's user avatar
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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 $\{ x_n \}$ ...
JCM's user avatar
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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 ...
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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 ...
Maximilian Janisch's user avatar
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69 views

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., $|...
ARNAB NANDI's user avatar
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56 views

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+\...
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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,\...
James Ilken's user avatar
12 votes
2 answers
887 views

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 $x>0$...
Satan's Minion's user avatar
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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 ...
Don's user avatar
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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, ...
psubodiosa's user avatar
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101 views

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_{...
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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 ...
user967210's user avatar
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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: $...
SnowRabbit's user avatar
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2 answers
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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)...
Jason Zheng's user avatar
6 votes
2 answers
636 views

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....
Tian Vlašić's user avatar
1 vote
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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 ...
Juan Moreno's user avatar
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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 ...
Anixx's user avatar
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5 votes
2 answers
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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, ...
Matěj Kudrna's user avatar
5 votes
1 answer
360 views

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 ...
yarchik's user avatar
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4 votes
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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}...
Todd Trimble's user avatar
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7 votes
1 answer
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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 ...
Daniele Tampieri's user avatar
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20 views

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}...
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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-...
alia's user avatar
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1 answer
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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^{\...
yannik0103's user avatar
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43 views

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 ...
stillconfused's user avatar
10 votes
1 answer
314 views

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 $...
Nilotpal Kanti Sinha's user avatar
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}...
Denys Lohvynov's user avatar
1 vote
0 answers
105 views

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 ...
Fei Cao's user avatar
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5 votes
1 answer
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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 ...
Ilya Zakharevich's user avatar
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0 answers
174 views

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} ...
Ricky's user avatar
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1 vote
0 answers
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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 ...
joy's user avatar
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22 votes
1 answer
4k views

A challenging (for me) limit calculation

How to calculate the following limit $$ \lim_{n\to\infty}\sqrt{n}\underbrace{{}\sin(\sin(\sin(\sin(\cdots\sin(\frac{1}{\sqrt{n}})\cdots))))}_{n \text{ sin's}} \text{?} $$ ${}{}$
C. WANG's user avatar
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0 answers
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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 $G_{\lambda}(x,p)$ ...
Davidi Cone's user avatar
5 votes
1 answer
447 views

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 ...
MathqA's user avatar
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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 ...
Pavel Gubkin's user avatar
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0 answers
102 views

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 ...
Max Muller's user avatar
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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 ...
Leo's user avatar
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4 votes
2 answers
408 views

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} \...
Alek Westover's user avatar
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 ...
Aadi Deepchand's user avatar
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}}\...
Staki42's user avatar
  • 151
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 $\...
Keen-ameteur's user avatar
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 ...
Dmytro Taranovsky's user avatar
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 ...
user1172131's user avatar
3 votes
0 answers
154 views

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 ...
Drew Brady's user avatar
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 ...
user1172131's user avatar
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}\}...
James Propp's user avatar
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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\...
Guoqing's user avatar
  • 431
2 votes
1 answer
692 views

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)}{\...
Steven Clark's user avatar
  • 1,061
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}.$...
Alek Westover's user avatar
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 ...
Mats Granvik's user avatar
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