# Questions tagged [asymptotics]

Asymptotic behavior of functions, asymptotic series and related topics

956
questions

0
votes

0
answers

22
views

### 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 ...

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_{\...

1
vote

0
answers

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
$$\frac{\pi}{4} = \sum_{k=1}^{n} c_{k} \arctan \left(\frac{1}{x_{k}}\right), \label{1}\tag{1} $$
...

0
votes

0
answers

103
views

### 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}+...

1
vote

1
answer

99
views

### 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)}\,...

1
vote

0
answers

40
views

### 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_{\...

2
votes

0
answers

52
views

### 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}{...

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 ...

1
vote

1
answer

245
views

### 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)...

4
votes

0
answers

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 ...

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 ...

5
votes

0
answers

152
views

### 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 ...

2
votes

0
answers

113
views

### 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
answers

44
views

### 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

381
views

### 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

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 ...

0
votes

0
answers

77
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., $|...

0
votes

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+\...

0
votes

0
answers

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,\...

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 $x>0$...

0
votes

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 ...

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, ...

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_{...

1
vote

0
answers

72
views

### 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

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:
$...

0
votes

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)...

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....

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 ...

-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 ...

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, ...

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 ...

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 ...

0
votes

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}...

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-...

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^{\...

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 ...

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 $...

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}...

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 ...

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 ...

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}
...

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 ...

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

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 $G_{\lambda}(x,p)$ ...

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 ...

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 ...

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 ...

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 ...

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} \...

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 ...