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Questions tagged [asymptotics]

Asymptotic behavior of functions, asymptotic series and related topics

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Heuristics for constrained maximal volumes in hypercubes as $n \to \infty$

It can be shown that there is a unique maximal surface of revolution with constant positive Gaussian curvature embedded in $[0,1]^3$ with a pair of antipodal points as cone points which attain the ...
53Demonslayer's user avatar
4 votes
1 answer
153 views

Uniform decay of $J'_{\nu}(x)$ for $x\gg1$

I need a uniform decay estimate for the derivative $J'_{\nu}(x)$ of the Bessel functions. By `uniform' I mean an estimate independent of $\nu$, at least for a range of orders like $\nu\ge0$. For $J_{\...
Piero D'Ancona's user avatar
1 vote
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88 views

Upper and lower bounds on the number of solutions to the equation $\frac{\pi}{4} = \sum_{k=1}^{n} c_{k} \arctan \left(\frac{1}{x_{k}} \right) $

Background The Norwegian mathematician and astronomer Carl Størmer did important work on the equation $$\frac{\pi}{4} = \sum_{k=1}^{n} c_{k} \arctan \left(\frac{1}{x_{k}}\right), \label{1}\tag{1} $$ ...
Max Muller's user avatar
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Sum with the fractional part function

Helo, I am still interested in the asymptotic behavior of certain sums of type $\sum_{k=1}^{n}\left\{ \frac{h(n)}{h(k)}\right\}$ and here I conjecture that we have $$\sum_{k=1}^{n}\left\{ \frac{2^{n}+...
 Babar's user avatar
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1 answer
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An asymptotic integral with complex phase

Suppose that $D\subset \mathbb R^2$ is the closed unit disk and that $f\in C^{\infty}(D)$. Assume that for all $\lambda \in (1,\infty)$ there holds $$ \left|\int_D f(x^1,x^2)\, e^{\lambda (x^1+ix^2)}\,...
Ali's user avatar
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Inequality with characteristic and moment generating functions

Suppose $\mu$ is a symmetric probability measure on $\mathbb{R}$. Define its characteristic and moment generating functions: \begin{align*} \phi_{\mu}(t) = \int_{\mathbb{R}}{\cos(xt) d\mu(x)} \\ M_{\...
Daniel Goc's user avatar
2 votes
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Are there any known Khinchin reals for which the asymptotics of "average" of their coefficients seems experimentally known?

We can define a Khinchin Real and recall the definition of Khinchin's Constant A real number $r$ is a Khinchin real if given the simple continued fraction expansion of $r$ as $$ r = a_0 + \cfrac{1}{...
Sidharth Ghoshal's user avatar
3 votes
0 answers
160 views

A sharper estimate for a generalization of the sum-of-divisors function

I am interested in the function $f_n(m)$ which can be defined by the Dirichlet generating function $$\zeta(s)\zeta(s - 1) \cdots\zeta(s - n + 1) = \sum\limits_{m = 1}^\infty \frac{f_n(m)}{m^s} $$ This ...
Bear's user avatar
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1 answer
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Decay estimate of energy minimizer and the linear ODE

Suppose $a$ is constant, $b(x)=C_b(1-x)$ and $c(x)=C_c(1+x^2)$ for some positive constants, we consider the minimizer of the energy $$E(u)=\int_{\mathbb{R}}\left[a^2(u'(x))^2/2+c(x)\left(\dfrac{1}{c(x)...
mnmn1993's user avatar
4 votes
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138 views

Uniform bound for Bessel functions

In NIST the following bound is claimed: $|J_\nu(x)|\le 1$ for all $x,\nu\ge0$. This is trivial for integer $\nu$, and it is pretty easy to prove a bound with 1 replaced, say, by 2. Does anyone have a ...
Piero D'Ancona's user avatar
4 votes
1 answer
243 views

Randomly removing length 1 intervals in an interval (a fragmentation process)

Short version: Start with a closed interval of length $t>0$ and repeatedly remove a random and uniformly distributed subinterval of length $1$ so long as this is possible. For $t$ large, what is ...
Gro-Tsen's user avatar
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A puzzle with magic Egyptian tilings

Background I've recently been devising a puzzle that incorporates elements from Egyptian fractions, magic squares, and tilings. The objective of the puzzle is to tessellate a square with sides of ...
Max Muller's user avatar
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2 votes
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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 ...
psubodiosa's user avatar
2 votes
0 answers
179 views

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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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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0 answers
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+\...
user avatar
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45 views

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
931 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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0 answers
74 views

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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3 votes
0 answers
100 views

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
2 votes
0 answers
107 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_{...
MikeG's user avatar
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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
2 votes
0 answers
142 views

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
129 views

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
648 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
0 answers
197 views

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
-1 votes
1 answer
164 views

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
  • 9,496
5 votes
2 answers
286 views

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
383 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
  • 482
7 votes
1 answer
464 views

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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0 answers
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}...
efsdfmo12's user avatar
  • 101
2 votes
1 answer
92 views

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
  • 23
0 votes
1 answer
153 views

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
0 votes
0 answers
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
321 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
193 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
116 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
  • 710
5 votes
1 answer
147 views

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
0 votes
0 answers
176 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
120 views

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
  • 119
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
  • 539
0 votes
0 answers
109 views

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
468 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
  • 313
2 votes
1 answer
200 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
1 vote
0 answers
106 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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1 vote
0 answers
89 views

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
  • 121
4 votes
2 answers
418 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
120 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

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