The asymptotics tag has no usage guidance.

**1**

vote

**1**answer

63 views

### Doubt concerning a sum involving Kummer extension degrees

I'd like to estimate the following sum
$$
\sum_{n\leq x}\frac1{k_n}\;,\qquad x\rightarrow \infty\;,
$$
where
$
k_n=[\mathbb{Q}(\zeta_n,a^{1/n}):\mathbb{Q}]
$
is the degree of a Kummer extension for a ...

**0**

votes

**0**answers

44 views

### Factorial Sums over Compositions or ``Unlabeled Permutations"

Let $C_n$ denote subset of integer compositions of $n$ and $c=(c_1,c_2,\dots c_n)$
In a divergent sum, the sequence
$$
a_n=\sum_{c\in C_n} \prod_{c_i\in c} c_i!
$$
frequently shows up and one ...

**2**

votes

**1**answer

72 views

### Asymptotic upper bound for recursive function $f(x) = f(x-1) + 2f\left(\lceil\frac{x}{2}\rceil \right)+2$

I'm looking for an asymptotic upper bound for the function f(x), which is recursively defined as follows: $$f(x) = f(x-1) + 2f\left(\lceil\frac{x}{2}\rceil \right)+2$$
with $f(1)=1$. I am pretty ...

**0**

votes

**0**answers

122 views

### Asymptotics to Taylor expansions?

I posted a question on MSE about approximating Taylor series but Despite a bounty I did not receive any answers or comments.
Maybe you guys can help.
...

**3**

votes

**1**answer

89 views

### Asymptotic Estimation of the Factorial Power Series

I want to estimate $f_p(e/p)$ as $p\to \infty$ for the series
$$
f_p(x) = \sum_{k=1}^p k! x^k
$$
I am sure this question in the right context is easy but I haven't found the information on how to ...

**3**

votes

**2**answers

137 views

### Asymptotics of solution of transcendental equation [closed]

It seems the real solution of the following equation $$t=u^2+u\log(u) $$ has no closed form in view of the output of
Reduce[t == u^2 + u*Log[u], u, Reals]
The ...

**2**

votes

**1**answer

82 views

### Rate of convergence of ergodic averages related to irrational rotation

Let $\alpha$ be an irrational number, consider the basic dynamical system $T^{n}(0) = \{n \alpha\}$ where $\{.\}$ denotes the fractional part.
Let $a < b$ be two numbers in $[0, 1]$. Then by ...

**1**

vote

**0**answers

27 views

### Derivatives of $O$-regular varying functions are $O$-regular varying functions?

The Monotone Density Theorem for regularly varying functions says, in essence:
Theorem (Monotone Density Theorem). Let $f$ be a differentiable regularly varying real-function of index $\rho$ ...

**4**

votes

**1**answer

101 views

### Asymptotic behaviour of an integral

For $k\in\mathbb{N}_{0}$ and $x\in\mathbb{R}$, define
$$I_{k}(x):=\int_{0}^{\pi/2}\cos(xg(\theta))\sin^{2k}\theta\,\mathrm{d}\theta$$
where
...

**6**

votes

**0**answers

239 views

### Asymptotics of A261668

In Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota-Baxter Algebras, Proposition 10.8, Jianqiang Zhao mentiones the sequence:
...

**-1**

votes

**1**answer

161 views

### Number of different factors of given size in primorial

Let $b_n$ be number of bits in product of all primes from $1$ to $n$ which is approximtely $b_n\approx n$.
What is the approximate number of distinct factors with number of bits ...

**1**

vote

**2**answers

106 views

### Asymptotic of the sum of squared primes [closed]

I have a rather simple question of number theory which I can't seem to be able to find a good reference for. I am not a specialist and I don't really know where to look. I would like to show that the ...

**4**

votes

**3**answers

152 views

### Asymptotic expression for $j$ which satisfies $\binom{n}{j}/j! \sim k$ as $n\to\infty$

Suppose $k>0$ is some fixed constant, and $n$ is a positive integer tending to infinity. Find $j\equiv j(n,k)$ such that
$$ \frac{\binom{n}{j}}{j!} \sim k. $$
The asymptotic expression for ...

**4**

votes

**1**answer

320 views

### Asymptotic $\int_M \mathrm{exp}[\mathbf{e}\left(n -\frac{t}{2\pi i}\right)] \left( 1 + \frac{5}{6} \mathbf{e}^2 \right)^{1/2} $ on quintic Calabi-Yau

Let $M = \{ G(x) = 0 \} \subseteq \mathbb{P}^4$ be a quintic Calabi-Yau and $\mathbf{e} \in H^2(M, \mathbb{Z})$ such that $\int_M \mathbf{e}^3 = 5$. Then as $t \gg 1$:
$$
\int_M
e^{n \mathbf{e}}
...

**5**

votes

**0**answers

88 views

### “Edge Density” of Infinite Planar Graphs

Given an infinite planar graph $G$, let's denote by $\{H_1,H_2,\dots,H_m\}$ all the labeled graphs on $n$ vertices that appear as subgraphs of $G$. Also let
$$d_n=\frac{\sum_{i=1}^m \#E(H_i)}{nm}$$
...

**29**

votes

**2**answers

930 views

### The coupon collector's earworm

[EDITED mostly to report on the answer by Kevin Costello
(and to improve the gp code at the end)]
I thank Nicolas Dupont for the following question
(and for permission to disseminate it further):
...

**7**

votes

**0**answers

142 views

### Counting the size of the largest sets of independent strings

This question derives from a PPCG coding challenge I posed previously but despite asking on math.se and offering a bounty, no progress has been made.
For a given positive integer $n$, consider all ...

**4**

votes

**0**answers

117 views

### A generalization of coupon collector problem - $\geq1$ pick per experiment

Mix $T\geq1$ coupons numbered $1$ to $T$ with a set of $S\geq0$ number of dummy coupons with no numbers. Select $N\geq1$ coupons at each trial at random and put them back.
$N=1$ is standard coupon ...

**3**

votes

**0**answers

134 views

### Table of LCM's vs. table of products

In 2004 Kevin Ford established sharp asymptotics on Erdős' problem on the number of different products $a\cdot b$, $a,b\in \{1,\dots,n\}$.
(http://arxiv.org/abs/math/0401223, see also discussion ...

**0**

votes

**0**answers

25 views

### Consistency of M-estimators when the constraint set also has to be estimated

Let $K \subset \mathbb R^n$ compact and convex. Also let $H$, $G_i, \; i \in \{1,\dotsc,m\} $: $K \to \mathbb R$ be convex functions.
Assume we have the following convex optimization problem:
$$
...

**2**

votes

**0**answers

121 views

### How distributive are the bad Laver tables?

Suppose that $n\in\omega\setminus\{0\}$. Then define $(S_{n},*)$ to be the algebra where $S_{n}=\{1,...,n\}$ and $*$ is the unique operation on $S_{n}$ where
$n*x=x$
$x*1=x+1\,\text{mod}\, n$ and
if ...

**0**

votes

**0**answers

66 views

### The Fransén-Robinson constant as a limit of integrals

This is the integral of the reciprocal gamma function,$$F=\int_0^\infty dx/\Gamma(x)= e+\int_0^\infty \frac{e^{-t}\ dt}{\log^2 t+\pi^2}$$ $$=e+\int_{-\infty}^{\infty}\frac{\exp(s-e^s)\ ...

**2**

votes

**1**answer

174 views

### Minimum distance between factorials and powers of 2

Let's define for a positive integer $n$: $$a(n) = \min \{|n! - 2^m| : m \in \mathbb N \}.$$ Does there exist a good asymptotic lower bound for the values $a(n)$ for large $n$? In particular, is the ...

**8**

votes

**3**answers

273 views

### Asymptotics of a recurrence relation

The sequence $(a_n)_{n \ge 0}$ satisfies, $a_0 = a_1 = 1$ and the recursion relation:
$$a_n = \sum\limits_{k=0}^{[n/2]} \frac{a_k}{(n-2k)!}$$
where, $[x]$ is the nearest integer to $x$ not exceeding ...

**2**

votes

**2**answers

118 views

### Reference request: using integral equations to study asymptotics of ODEs

I was told by my supervisor that one way to study the asymptotic behaviour of solutions to ODEs is "to reformulate them as integral equations, and use fixed-point kind theorems on the resulting ...

**5**

votes

**0**answers

190 views

### Paths in Pascal's triangle; or balanced $0-1$ initial segments

Here is a problem arising (via a tortuous path) from trying to determine the spectrum of Vershik's adic map on Pascal's triangle (a moderately well-known question: is the spectrum trivial, that is, is ...

**2**

votes

**1**answer

117 views

### Bounding $\int_{\mathbb{R}_{+}}W(x,\lambda)W(y,\lambda)d\lambda$

Let $$W:\mathbb{R}_{+}^{2}\rightarrow(0,1]$$ be a symmetric, integrable function. Let $$f(x)=\int_{\mathbb{R}_{+}}W(x,\lambda)d\lambda$$ and assume this function is monotonically non-increasing. Is ...

**2**

votes

**1**answer

89 views

### Asymptotic expansion of a sequence given by an integral with reciprocal Gamma function

I would like to know the asymptotic expansion of the sequence of positive numbers given by
$$I_{n}:=-\int_{0}^{1}\frac{n^{x-1}}{\Gamma(x-1)}dx,$$
for $n\rightarrow\infty$.
One can easily derive an ...

**13**

votes

**4**answers

398 views

### Maximum of the Vandermonde determinant / minimum of the logarithmic energy

The problem is to find the asymptotics (as $n\to\infty$) of the maximum (say $M_n$) of the Vandermonde determinant
$$V_n:=\prod_{0\le i<j\le n-1}(a_j-a_i)
$$
over all $a_0,\dots,a_{n-1}$ such ...

**0**

votes

**0**answers

15 views

### Question on Asymptotic Normality of non-parameter estimands of a distribution

I'm currently taking an introductory statistics course and one of the topics we covered was Maximum Likelihood Estimates and their asymptotic normality (under reasonable conditions that were not ...

**2**

votes

**0**answers

217 views

### Hamming weight probability of projections

Given $s,t\in(0,1)$, $c>1$, $n\in\Bbb N$, pick $2^{n^t}$ random vectors $\{v_i\}_{i=1}^{2^{n^t}}$ such that each $v_i\in\{x\in\{0,1\}^{2^n}:|x|_{hamming}={2^{n-n^s}}\}$.
If $v_i^\perp$ is ...

**1**

vote

**0**answers

58 views

### Related to derivative of Modified Bessel I function wrt the order

I recently met some problems related to the modified Bessel I funtions. Let $I(\nu,x):=I_\nu(x)$, and $I'_\nu(\nu,x):=\dfrac{\partial}{\partial \nu}I(\nu,x)$.
Using maple, it seems that ...

**1**

vote

**0**answers

45 views

### help with an asymptotic estimate for a certain product

(I apologize in advance if this question is not suitable for Math Overflow, but it came up in a research problem and thought perhaps I could find some help here.)
I'm having difficulty finding an ...

**2**

votes

**0**answers

47 views

### Functions whose Laplace transforms have prescribed behavior at minus infinity

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a non-negative function with entire Laplace transform $\hat{f}$ (in particular $\lim_{t\to \infty}e^{st}f(t)=0$ for all $s$), and $p_0$ a positive integer. ...

**3**

votes

**1**answer

115 views

### Reference of $\hbar$-differential operator from symplectic geometry perspective

I am reading Bates and Weinstein's book 'Lectures on the Geometry of Quantization'. In Chapter 6, they defined the $\hbar$-differential operator, and showed (Theorem 6.7) that the Lagrangian ...

**5**

votes

**1**answer

218 views

### The growth rate of $\left(1+\frac{x}{f(x)}\right)^{f(x)}$

I was reading this question on functions "in the middle" of linear and exponential growth, and it caused me to think of this function:
$$ G(x) = \left(1 + \frac{x}{f(x)}\right)^{f(x)} $$
for some ...

**8**

votes

**2**answers

294 views

### Tauberian theorem with better error term

This is a fairly vague question.
Suppose we have a sequence of positive numbers $(c_n)_n$ and we want to find an asymptotic formula for $S(x) = \sum_{n \leq X} c_n$. In favorable circumstances, ...

**2**

votes

**2**answers

188 views

### Estimating a Selberg-type integral (or a Fredholm determinant)

I am concerned with the asymptotical behavior of integrals like this for large $n$
$$\frac{1}{n!}\intop_{\Omega}\prod_{1\leq i<j\leq n}(x_{j}-x_{i})^{2}\,\prod_{j=1}^{n}e^{-x_{j}^{2}}dx_{j},$$
...

**6**

votes

**1**answer

168 views

### Upper bound for a Selberg-type integral over a rectangular region

(Cross-posted from math-SE).
I am trying to estimate the values of the following integral for large $n$,
$$\frac{1}{n!}\intop_{\Omega}\prod_{1\leq i<j\leq ...

**2**

votes

**1**answer

75 views

### Asymptotic expansion of a Laplace-type integral with a “manifold of maxima”

Moved this here from MathSE because I had no luck there and suspect the question may be harder than I first thought.
Consider the integral
$$
I(\alpha)=\int_0^1 dx_1 \int_0^1 ...

**9**

votes

**2**answers

375 views

### Expected centered entropy of the binomial distribution

In short, the function I am interested in is the following:
$$I_n(p) = \sum_{k=0}^n \binom{n}{k} p^k (1-p)^{n-k} \left[h(p) - h\left(\tfrac{k}{n}\right)\right],$$
where $h(x) \triangleq -x \log x - ...

**9**

votes

**5**answers

780 views

### Combinatorial\Probabilistic Proof of Stirling's Approximation

Stirling's approximation is the following well-known asymptotic result:
$$n! \approx \left(\frac{n}{e}\right)^n \sqrt{2 \pi n}$$
This result has several analytical proofs, for example via Laplace's ...

**6**

votes

**1**answer

174 views

### When is this sum of perfect powers bounded

For any positive integers $n,d$, let
$$
A_d(n)=\frac{\sum_{k=1}^n k^{2d}}{n(n+1)(2n+1)}
$$
It is easy to see (and well-known) that for fixed $d$, $A_d(.)$ is
a polynomial of degree $2d-2$. Then we ...

**1**

vote

**1**answer

202 views

### Asymptotic of a certain double sum involving binomial coefficients

Consider sums of the form
$S(n)=\sum^{n}_{m=0}\sum^{m}_{k=1}2^{2k+m+1}{n-m+k+1 \choose 2k+2}{m \choose k}$
I am interested in the asymptotics of $S(n)$ as $n\to \infty$.
More precisely I would ...

**6**

votes

**3**answers

353 views

### A balls and urns model for a hashing problem

Fix $N \in \mathbb{N}$. Suppose we throw $N$ numbered balls into $N$ numbered urns, so that for each $b \in \{1,\ldots,N\}$, ball $b$ lands in urn $j$ with equal probability $1/N$. Choose a number $c ...

**6**

votes

**2**answers

305 views

### Asymptotic expansion of $\zeta(s \mid a,b)= \sum_{n=1}^{\infty} \frac{1}{(n+a)^{s}(n+b)}$

I'm interested in an asymptotic expansion of the following Riemann zeta-type function
$$
\begin{align}
\displaystyle \zeta(s \mid a,b) := \sum_{n=1}^{\infty} \frac{1}{(n+a)^{s}(n+b)},
\quad \Re a ...

**0**

votes

**1**answer

159 views

### Proving a complicated inequality with powers of logarithms

I am currently formalising some results from complexity theory with a theorem prover. For that, I have to prove the following statement:
Let $p, b, \varepsilon \in \mathbb R$ with $\varepsilon>0$ ...

**0**

votes

**0**answers

90 views

### Condition for boundedness in stationary phase theorem

I am trying to understand theorem 7.7.1 in Hormander's Analysis of linear partial differential operators, vol.1.
Let $K \subset \mathbb{R}^n$ be a compact set, $X$ an open neighborhood of $K$ and ...

**1**

vote

**1**answer

141 views

### Infinite sum of asymptotic expansions

I have a question about an infinite sum of asymptotic expansions:
Assume that $f_k(x)\sim a_{0k}+\dfrac{a_{1k}}{x}+\dfrac{a_{2k}}{x^2}+\cdots$
with $a_{0k}\leq \dfrac{1}{k^2}$, $a_{1k}\leq ...

**1**

vote

**0**answers

103 views

### Number of digits in $n$-th term of generalized Fibonacci/Narayana sequence

Let $d$ be a nonnegative integer, and let the sequence ${F_d(n)}$ be defined as follows:
$F_d(n) = 1$ for $n = 0, 1, \ldots, d$
$F_d(n) = F_d(n-1) + F_d(n-1-d)$ for $n>d$
For $d=0$ the sequence ...