45
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 ...
- 1,555
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 ...
- 172k
19
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 ...
- 3,120
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&...
- 84.5k
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
$$ f^{(2)}(x) = \sum_{j=0}^{q-3} \frac{f^{(2+j)}(0)}{j!} x^j + O( \frac{(\ln q)^{-q} |x|^{q-2}}{(...
- 106k
16
votes
A new way of approaching the pole of the Riemann zeta function - and a new conjectured formula
The first formula is trivial. $$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)^...
- 3,315
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 $X$ ...
- 13.8k
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 ...
- 13.4k
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 ...
- 131k
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-...
- 110k
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\{...
- 172k
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
$$\sum_{n=0}^{\infty}e^{-2^n/x} = \frac{1}{2\pi i}\int_{3-i\infty}^{3+i\infty}\frac{2^s}{2^...
- 5,014
12
votes
Accepted
Asymptotics of a recurrence relation
It is not hard prove the bounds you want by purely real variable techniques. First note that the $a_n$ are non-negative for all $n$. For a general non-negative sequence $a_n$, and real numbers $N>...
- 43.2k
12
votes
Accepted
Asymptotics of product of Euler's totient function (A001088)?
We may just write down $\varphi(k)=k\cdot \prod_{p|k} (1-1/p)$, $p$ runs over the primes which divide $k$, then
$$
\frac{\left(\prod_{k\leqslant n}\varphi(k)\right)^{1/n}}{n}=\frac{\sqrt[n]{n!}}n\...
- 98.7k
12
votes
Accepted
Asymptotics of a special function
Gradshteyn & Ryzhik equation 3.773.1 gives (for $q>0$)
$$\frac{1}{q}B(q)=\int_0^\infty \frac{\sin x}{x(q^2/4+x^2)^{1/2}}\,dx=\frac{1}{q}G^{21}_{13}\left(\frac{q^2}{16}\biggl|^1_{1/2,1/2,0}\...
- 172k
12
votes
Accepted
About the logarithmic derivative of the Riemann zeta function
I think your final goal follows by taking the logarithmic derivative of the functional equation:
$$\frac{\zeta'}{\zeta}(s)+\frac{\zeta'}{\zeta}(1-s)=\log\pi-\frac{1}{2}\frac{\Gamma'}{\Gamma}\left(\...
- 95k
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 ...
- 95k
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(...
- 22.6k
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
$$\psi(x)-x=\Omega_{\pm}(x^{1/...
- 95k
12
votes
Slick proof of Stirling's Formula?
The proof in the OP based on the sequence $a_n$ is proof number 1 in Steve Dunbar's Dozen Proofs of Stirling’s Formula (page 8, worked out here). Is there an alternative proof based on a sequence $b_n$...
- 172k
12
votes
Accepted
Prove or disprove that $\sup_{n\in\mathbb{N}}\left|\sum_{\substack{d|n \\d<Q}}\mu(d)\right|\sim\pi(Q)$
This is not true. In fact
$$
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}.
$$
The upper bound is due to Montgomery and Vaughan (see ...
- 43.2k
12
votes
Accepted
Is there a real-analytic way to derive the asymptotics of $\int_{-\infty}^\infty e^{ikx} e^{-k^4}\,dk$ as $|x|\to\infty$?
A differential equation for ${\cal A} (x) $ can be obtained as follows,
$$
\frac{d^3}{dx^3 } {\cal A} (x) = \int_{-\infty }^{\infty } dk\, (-ik^3 ) e^{ikx} e^{-k^4 } = \frac{x}{4} \int_{-\infty }^{\...
- 5,267
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 $...
- 98.7k
12
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, ...
- 110k
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 ...
- 172k
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 ...
- 176
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 ...
- 131k
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 ...
- 3,143
11
votes
Accepted
Asymptotic of integral $\int_{1}^{e^n}(1-\frac{\ln(x)}{n})^n\,dx$
Denote $t=\ln(x)/n$, then $t$ varies from 0 to 1 and the the integral reads as $$n\int_0^1 ((1-t) e^{t})^ndt.$$
We have $(1-t)e^t=(1-t)(1+t+t^2/2+\ldots)=1-t^2/2+O(t^3)=\exp(-t^2/2+O(t^3))$ for small $...
- 98.7k
11
votes
Accepted
Asymptotics for $\prod(1-\frac{1}{p})$ over all primes $p\leq x$ with $p \equiv 3 \bmod 4$
I assume that you meant to write product and not sum. Defining
\begin{equation*}
\small D(x)=\prod_{\substack{p\leq x\\ p\equiv 1 \bmod 4}} \Big(1+\frac{1}{p}\Big) \mbox{ } \mbox{ }
\end{equation*}...
- 2,374
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