41
votes

Accepted

### Wrong asymptotics of OEIS A000607 (number of partitions of an integer in prime parts)?

Your data is compatible with the more refined estimates proved by Vaughan in Ramanujan J. 15 (2008), 109–121. His Theorems 1 and 2 (together with his (1.9)) reveal that
$$\log(A000607(n)) = 2 \pi \...

- 87k

22
votes

Accepted

### The coupon collector's earworm

For the asymptotic case: Let $t_1=n \log n - Cn$ and $t_2 = n \log n + Cn$, where $C$ is slowly tending to infinity. It is a classic result that as $C$ tends to infinity the probability all coupons ...

- 5,571

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 ...

- 153k

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 ...

- 2,945

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)^...

- 2,979

16
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&...

- 82.5k

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 $X$ ...

- 12.9k

15
votes

Accepted

### Cubic-exponential enumerative combinatorics

Since you mentioned Cayley's theorem for spanning trees, I believe one important example is its higher dimensional analogue due to Kalai. Indeed the number of simplicial spanning trees of the k-...

- 82.5k

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 ...

- 119k

14
votes

Accepted

### Probability that a positive integer is the euler phi function of another positive integer

See Erick Wong's response here. In particular, Kevin Ford proved (in more precise form) that
$$ f(n) = \frac{n}{\log n} \exp\left(O(\log \log \log n)^2\right),$$
whence $f(n)/n$ tends to zero. The ...

- 87k

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\{...

- 153k

13
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-...

- 82.5k

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\...

- 90.4k

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}\...

- 153k

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(\...

- 87k

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 ...

- 87k

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(...

- 21.8k

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/...

- 87k

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 ...

- 42.6k

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 }^{\...

- 4,386

11
votes

Accepted

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

Write $V(a)$ for the determinant $\prod_{0\leq i<j\leq n-1} |a_i-a_j|$. Selberg's formula tells you that
$$\int_0^1 \cdots \int_0^1 V(a)^{2\beta} \prod_{i=0}^{n-1} da_i=
n! \prod_{j=0}^{n-1} \frac{...

- 7,154

11
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>...

- 42.6k

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 ...

- 119k

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 ...

- 3,023

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 $...

- 90.4k

11
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 ...

- 40.3k

11
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 $...

- 90.4k

11
votes

### Asymptotics for $\int\exp( -x t / \log t)dt$

Take any real $a>e$. Then
\begin{equation*}
f(x)=f_{1,a}(x)+f_{2,a}(x), \tag{1}\label{1}
\end{equation*}
where
\begin{equation*}
f_{1,a}(x):=\int_e^a dt\,e^{-xg(t)},\quad
f_{2,a}(x):=\...

- 82.5k

10
votes

Accepted

### Bound on sum of complex summands involving binomial coefficients

Assuming that $|x+y|<1$ and $4|xy| \le 1$, here's a proof of the decay.
First suppose that $|x|> |y|$. The desired sum is
$$
\le \binom{2n}{n} |xy|^n \sum_{j=0}^{n} |y/x|^j \le
\binom{2n}{...

- 42.6k

10
votes

### Asymptotics of a recurrence relation

Let $A(x)$ denote the formal generating function of $\{ a_n\}$. The recurrence relation can be written as $A(x)=e^x A(x^2)$. Applying this repeatedly, we find $A(x) = e^x e^{x^2} \cdots = e^{f(x)}$, ...

- 9,364

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