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 ...

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 ...

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 ...

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&...

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}}{(...

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)^...

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 $X$ ...

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 ...

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 ...

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-...

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\{...

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^...

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>...

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\...

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}\...

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(\...

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 ...

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(...

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/...

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 $b_n$...

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 ...

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 }^{\...

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 $...

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, ...

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 ...

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 ...

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 ...

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 ...

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 $...

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*}...

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