46
votes
Accepted
A challenging (for me) limit calculation
This limit converges to $\frac{\sqrt3}2$. The idea is that $\sin(x) = x - \frac{x^3}6 + O(x^5)$, so we start with $\frac1{\sqrt n}$ and repeatedly subtract $\frac{x^3}6$. We can approximate this ...
- 3,319
22
votes
A new way of approaching the pole of the Riemann zeta function - and a new conjectured formula
The paper entitled Euler constant as a renormalized value of Riemann zeta function at its pole by Andrei Vieru contains a derivation of the first formula in the OP (Benoȋt Cloitre's formula), and ...
- 188k
21
votes
Accepted
Asymptotic expansion of $\sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$
I sketch the arguments for $C(x)$, the arguments for $L(x)$ are essentially the same.
The specific form of the sum suggests probabilistic arguments.
Let $X_x$ be a $\mathrm{Poiss}(x^2)$-distributed ...
- 3,255
17
votes
Accepted
How does the number of trees on $n$ vertices *up to isomorphism* grow as $n \to \infty$?
For Q1 the answer is known to be $\sim C_1C_2^n n^{-5/2} $ for $C_1\approx 0.5349496061...$ and $C_2\approx 2.9955765856...$. This can be found in Flajolet and Sedgewick's "Analytic Combinatorics&...
- 85.5k
17
votes
Accepted
Is there a nonpolynomial $C^\infty$ function $f$ such that $\sup_{x \in \mathbb{R}} \lvert f^{(q)}(x) \rvert \leq (\ln q)^{-q}$ for every $q >1$?
The answer is no. From Taylor's theorem with remainder, we see that for any integer $q>2$, we have
$$ f^{(2)}(x) = \sum_{j=0}^{q-3} \frac{f^{(2+j)}(0)}{j!} x^j + O( \frac{(\ln q)^{-q} |x|^{q-2}}{(...
- 114k
16
votes
A new way of approaching the pole of the Riemann zeta function - and a new conjectured formula
The first formula is trivial. $$f(s)= \frac1{s-1}+\gamma +O(s-1)$$ $$g(z)=1+2^{-z}+3^{-z}+4^{-z}+O(5^{-z})=1+2^{-z}(1+(3/2)^{-z}+(4/2)^{-z}+O(5/2)^{-z})$$
$$f(g(z)) = \frac1{2^{-z}(1+(3/2)^{-z}+(4/2)^...
- 3,403
16
votes
Accepted
Density of the set of numbers whose sum of digits is prime
Yes, $A(n)$ has zero natural density. It suffices to prove this for $n$ which is a power of $10$.
and it is possible to make this more precise. To see this, first let $n=10^k$ and note that for $X$ ...
- 14.2k
16
votes
Accepted
Mertens-like theorem
This lies beyond Mertens, in the sense that this variant actually implies the Prime Number Theorem, as will be explained below, while Mertens' theorem is weaker than the PNT.
I sketch below a complex ...
- 14.6k
15
votes
Accepted
Distinct exponents in the factorization of the factorial, a problem of Erdős
The primes $p \leq \sqrt{n}$ can be ignored as the number of them is $$\approx \sqrt{n} / \log (\sqrt{n} )= 2 \sqrt{n}/\log n = o (1) \cdot \sqrt { n/\log n}$$
For primes $p> \sqrt{n}$, the ...
- 148k
14
votes
Accepted
What is (approximately) the expected value of $X\log{ X}$ where $X$ is binomial (or Poisson)?
$\newcommand{\ep}{\varepsilon}
$
Let $X$ be any nonnegative random variable (r.v.) with finite mean $\mu>0$ and variance $\sigma^2<\infty$. For any real $u>0$, we have $\ln\frac xu\le\frac xu-...
- 128k
14
votes
Accepted
Asymptotic behavior of $\sum_{k=1}^{\infty} \sqrt{\max\{1 - k^2/x^2,0\}}$ as $x\to\infty$
Use the Euler-MacLaurin formula,
$$\sum_{k=1}^\infty F(k)=\int_0^\infty F(k)\,dk+\tfrac{1}{2}[F(\infty)-F(0)]+\int_0^\infty (k-\text{Int}\,[k]-\tfrac{1}{2})F'(k)\,dk.$$
In this case $F(k)=\sqrt{\max\{...
- 188k
14
votes
Accepted
Asymptotics of a strange oscillatory function
I will show (unless I made a mistake)
$$
f(x) = a\sqrt{x} + O(x^{1/3}),
$$
for some constant $a > 0$.
For each $x > 0$, let $g(t) = \sin(x/t^2)$. Let $\varepsilon > 0$ to be chosen later ...
- 4,661
13
votes
Accepted
How to determine the asymptotics of $\sum_{n=0}^{\infty} e^{-\frac{2^n}{x}}$
Note that
$$\sum_{n=0}^{\infty}\frac{1}{(2^n)^s}=\frac{2^s}{2^s-1}.$$
It now follows by Mellin inversion that
$$\sum_{n=0}^{\infty}e^{-2^n/x} = \frac{1}{2\pi i}\int_{3-i\infty}^{3+i\infty}\frac{2^s}{2^...
- 5,089
13
votes
Accepted
Asymptotics of $\int_0^\infty \frac{x^{2z}}{\Gamma(1+z)}\,dz$ for large $x$
$\newcommand\Ga\Gamma$Let $g(x)$ denote your integral. Then
$$g(x)\sim e^{x^2} \tag{1}\label{1} $$
as $x\to\infty$.
Proof:
$$g(x)=\int_0^\infty dz\,\frac{x^{2z}}{\Ga(1+z)}. \tag{1.5}\label{1.5} $$
So, ...
- 128k
13
votes
Non-atomic probability measures on N
You may want to look at "Finitely Additive Measures on the Integers" by Dorothy Maharam. Note in particular that finitely additive positive measures on $\mathbb{N}$ are the same as countably ...
- 10.3k
12
votes
Accepted
The number of representations of an integer as the inner product of integral lattice points
The asymptotic formula is true for even dimensions $k\geq 2$. We can prove this by induction on $k$, inspired by Rodrigo's observation on Eisenstein series.
The case $k=2$ is classical and addressed ...
- 105k
12
votes
Accepted
On approximation of $\sum_{a,b=1}^n\gcd(a,b)$
The asymptotic you want does not hold just because the "last-term fluctuation"
$$ g(n)-g(n-1) = 2\sum_{a=1}^n \gcd(a,n)-n $$
is too large. Indeed, denoting the sum in the right-hand side by $\sigma(...
- 23k
12
votes
What is the asymptotic of the irregular blue curve? Is it $(8x)^{1/2}$ or is it something else?
Let us denote the left hand side of $(1)$ by $\psi(x)$. It is known that $|\psi(x)-x|$ is not bounded by a constant times $x^{1/2}$. In fact Littlewood (1914) proved that
$$\psi(x)-x=\Omega_{\pm}(x^{1/...
- 105k
12
votes
Slick proof of Stirling's Formula?
The proof in the OP based on the sequence $a_n$ is proof number 1 in Steve Dunbar's Dozen Proofs of Stirling’s Formula (page 8, worked out here). Is there an alternative proof based on a sequence $b_n$...
- 188k
12
votes
Accepted
Prove or disprove that $\sup_{n\in\mathbb{N}}\left|\sum_{\substack{d|n \\d<Q}}\mu(d)\right|\sim\pi(Q)$
This is not true. In fact
$$
x(\log x)^{-1+1/\pi} \gg \sup_n \Big| \sum_{\substack{ d|n \\ d\le x}} \mu(d) \Big| \gg x (\log x)^{-1+1/\pi}.
$$
The upper bound is due to Montgomery and Vaughan (see ...
- 43.7k
12
votes
Accepted
Is there a real-analytic way to derive the asymptotics of $\int_{-\infty}^\infty e^{ikx} e^{-k^4}\,dk$ as $|x|\to\infty$?
A differential equation for ${\cal A} (x) $ can be obtained as follows,
$$
\frac{d^3}{dx^3 } {\cal A} (x) = \int_{-\infty }^{\infty } dk\, (-ik^3 ) e^{ikx} e^{-k^4 } = \frac{x}{4} \int_{-\infty }^{\...
- 6,696
12
votes
Accepted
How hard is it to say "not exactly $p$" with a Horn sentence?
I will show how to improve Tulipani’s construction from $O(p^5)$ symbols to $O(p^3)$ symbols, or $O(p^3\log p)$ bits.
Recall that Tulipani’s sentence is
$$H_p=\forall\vec x\,\exists\vec s\,\exists\vec ...
- 47.1k
12
votes
Accepted
Asymptotics for $\int\exp( -x t / \log t)dt$
Denote $t/\log t=y$. Then $y$ increases from $e$ to $\infty$ when $x$ goes from $e$ to $\infty$, and $dt(1/\log t-1/\log^2 t)=dy$, thus $dt\sim \log t\cdot dy\sim \log y\cdot dy$ for large $t$ (since $...
- 109k
12
votes
Asymptotic behavior of a certain oscillatory integral
$$I(x):=\int_{0}^{\infty}\frac{e^{i r}}{r^{\frac{1}{2}}}\int_{0}^{\infty}\frac{e^{-s}}{s^{\frac{1}{2}}}\frac{r}{sx+\sqrt{sxr}+r}ds dr.$$
To aid the asymptotic analysis, I regularize $I(x)$ by ...
- 188k
12
votes
Accepted
Asymptotics for pairs of positive integers whose harmonic (resp. geometric) mean is an integer
[CORRECTED 8/28]
I think the asymptotic ratio is $(4/3)*\ln(2)\approx 0.924196$.
$$G(n) = (6/\pi^2)*n*\ln(n) + O(n)$$
$$H(n) = (8\ln(2)/\pi^2)*n*\ln(n) + O(n)$$
The proof for the geometric mean goes ...
- 316
12
votes
Accepted
Lindelöf hypotheses for derivatives of zeta
The Lindelöf hypothesis yields the same bound for each derivative of $\zeta(s)$ via Cauchy's formula. Indeed, let $n\in\mathbb{N}$, $\sigma\in\mathbb{R}$, $T\in(1,\infty)$, $\varepsilon\in(0,1/2)$. ...
- 105k
12
votes
Accepted
Sum of reciprocals of rough numbers
(I am going to switch your $x$ and $y$ for aesthetic purposes.)
The correct upper bound is $\log x/\log y$, which in particular is bounded if $y$ is a power of $x$.
Let $\Phi(x,y)$ be the number of $y$...
- 14.6k
11
votes
Accepted
For a round-robin tournament, what is the favorite's least favorite size?
I can show that $N(\epsilon)$ is equal to $\epsilon^{-2}$ up to a log factor on each side.
The strategy I'll use is to give an upper bound for $\pi(1/2+\epsilon,n)$. Optimizing it, we obtain an upper ...
- 148k
11
votes
How did they come up with the MRRW bound?
For linear programming type bounds it is sometimes only possible to give effective bounds (that is bounds that work and are manageable) and what is surprising is that the primitive method often gives ...
- 3,209
11
votes
Accepted
Differentiating an integral that grows like log asymptotically
The answer is no, even in the smooth case. Take for example:
$$
f(x) = \frac{2}{x} + \frac{\cos(\log(x))}{x}
$$
Alter it on a small neighborhood of $0$ in such a way that there is no singularity ...
- 3,223
Only top scored, non community-wiki answers of a minimum length are eligible