22
votes

Accepted

### Freeman Dyson's approach to string theory

Dyson's A walk through Ramanujan's garden gives the background of this comment: He explains that the "seeds from Ramanujan's garden have been
blowing on the wind and have been sprouting all over ...

10
votes

### Freeman Dyson's approach to string theory

I don't think it would have convinced Feynman because he didn't like the rabbit hole that string theory seemed to be going down. That instead of trying to explain some phenomenon, that they were ...

9
votes

Accepted

### Is there a Cramer's conjecture for Sophie Germain primes?

Heuristics says that $n$ is Sophie Germain prime with probability roughly $1/\log^2n$. Thus the probability that $C\log^an$×consecutive numbers starting from $n$×are not Sophie Germain primes is about ...

9
votes

### Is there an Erdős–Kac theorem for number of divisors?

Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Chapter III.4:
The frequencies of the number of distinct prime divisors $\omega(n)$ of $n$ obey:
$$
\frac{\#\{1\leq n \leq N:\omega(...

9
votes

### If normal with respect to prime base then normal for all bases

Gerry Myerson already answered in the comments, but let me be a little more explicit. Take any number $m\in\mathbb{N}$ with at least two prime factors (for example $6$). Then construct a Cantor set $C=...

7
votes

### If normal with respect to prime base then normal for all bases

Wolfgang Schmidt, Über die Normalität von Zahlen zu verschiedenen Basen, Acta Arith. 7 1961/1962 299–309, MR0140482 (25 #3902), according to the review by N. G. de Bruijn, proved this: Let the set of ...

7
votes

Accepted

### Small linear relations between primitive Pythagorean triples $\mathsf{II}$

Yes, the minimal $\|(u,v,z)\|_\infty$ is within a constant factor of
$\sqrt{|c|}$ (equivalently, of $\sqrt{\max(|a|,|b|)}$.
The orthogonal complement of $(a,b,c) = (m^2-n^2, 2mn, m^2+n^2)$
contains ...

6
votes

Accepted

### Almost evenly distributed spherical random vectors

$\newcommand\PP{\mathbb P}$
Surely $n_\min \lesssim d$, because it works for $c = 1/4$ and $n=160d$.
We use that the number of "distinct" $v$ with respect to the classifiers $\textrm{sgn}\...

4
votes

Accepted

### How many roots of polynomial in $\mathbb Z[x]$ and $\mathbb Q[x]$ are integers on average?

$$\mathbb{E}[\text{#|roots of P|}]=\mathbb{E}(\sum_{k\in \mathbb{Z}}1_{k \text{ is a root of P}})=\sum_{k\in\mathbb{Z}}\mathbb{P}(P(k)=0)$$
For $k=0$, $\mathbb{P}(P(0)=0)=\frac{1}{(2B+1)}$.
For $k\...

4
votes

Accepted

### Non-asymptotic results in probabilistic number theory

Chebyshev's bias says that there are slightly more non-Pythagorean primes than Pythagorean primes (although the limiting frequency is the same).

4
votes

Accepted

### Small linear relations between primitive Pythagorean triples $\mathsf I$

Yes, when $m>n>0$ and
$$ a = m^2 - n^2 $$
$$ b = 2mn $$
$$ c = m^2 + n^2 $$
then
$$ -n a^2 +(m-n)b^2 - n ab +(m-n)bc - n ca = 0 $$
or quintuple
$$ -n, m-n, -n, m-n, -n $$
There is a second ...

4
votes

Accepted

### Why is $\sum_{m=1}^{n}\frac{(\nu(m)-\log\log n)^2}{n\log\log n}=\int_{-\infty}^{\infty}\omega^2\, \mathrm{d}\sigma_n(\omega)$?

Denote $f_m=(\nu(m)-\log\log n)(\log\log n)^{-1/2}$ and define $\rho(\omega)=\sum_{m=1}^n\delta(\omega-f_m)$, with $\delta(x)$ the Dirac delta function. Because of the identity $\int g(\omega)\delta(\...

3
votes

### Asymptotic density of an infinite union of subgroups

It's false. Up to an (irrelevant) factor of $2$, I work with $\mathbb{N}$ instead of $\mathbb{Z}$.
The only ingredient needed is that $c_D \to 0$ as $D \to \infty$, where $c_D > 0$ is such that $$\...

3
votes

Accepted

### Is $\lim_{s \rightarrow 1} E(f(X_s)) = \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{k=1}^N f(k)$?

The answer is no. E.g., if
$$f(k)=\sum_{j=0}^\infty 1(2^{2j}<k\le2^{2j+1})$$
for natural $k$, then
\begin{equation}
Ef(X_s)\to\frac13 \tag{1}\label{1}
\end{equation}
as $s\downarrow0$, whereas
$...

3
votes

### Ways of proving normal distribution (with a view towards Selberg's central limit theorem)

There is also the information-theoretic method; see Linnik (1959), Barron (1986), etc. Oliver Johnson has written an entire book on this topic.

3
votes

### Is there a connection of prime numbers and extreme value theory?

This answer is maybe two years too late, but better too late then not answered:
Let $N \ge 3$ be a natural number. Consider the set $\Omega_N:= \{y \in \mathbb{N}| 1 < y \le N, \gcd(y,N) =1 \}$.
...

2
votes

Accepted

### Approximately satisfying simultaneous vector linear diophantine equations?

Even allowing $A$ and $B$ to be real numbers, the vectors $A a + B b$ will all lie in some fixed plane $P$. But then, if $n \ge 3$, for all but $\epsilon$ of the possible values of $c$ one will have
$...

2
votes

### Upper bound for an infinite series of Pochhammer Symbol

The sum is $r/(1-\alpha)^{(1+r/\alpha)}$ by the binomial theorem.

2
votes

### How differently would we model the distribution of primes if prime gap is larger?

I think a short answer is we don't have a supply of alternative models for prime numbers lying around that make very different predictions for gaps.
If a proof was found of larger prime gaps than ...

2
votes

### A variant of Turán–Kubilius inequality

For any real $A \neq 1$ we have
$$
\sum_{n \leq x} (\omega(n) - A \log \log x)^2 \sim (1-A)^2 x (\log \log x)^2
$$
as $x \to \infty$. This is a consequence of the quoted result
$$
\sum_{n \leq x} (\...

1
vote

### A variant of Turán–Kubilius inequality

Related:
R.L. Duncan, Some Applications of the Turan-Kubilius Inequality in Proceedings of the American Mathematical Society 30(1), September 1971, has proved the following:
Let $g:\mathbb{N}\...

1
vote

### Probabilistic interpretation of square free numbers and other properties

The standard way to formalize this thought is via the concept of "density".
This is because you of course cannot uniformly randomly select $n\in\mathbb{N}$, but you can instead examine (for a subset $...

1
vote

### Approximately satisfying simultaneous vector linear diophantine equations?

For $n=1$ the probability that such $A$, $B$ exist is at least
$$\frac6{\pi^2}\left(1+\frac18\right)=68\%$$
since it can happen at least in the following two disjoint ways:
$a$ and $b$ coprime
$a$, $...

1
vote

Accepted

### On distribution of size of integer points in a subspace associated to a linear diophantine equation

We will assume that $n<A,B,C,D<2n$ and $\gcd(A,B)=\gcd(C,D)=1$.
For $x,y,z\in\mathbb{Q}$, we write $v(x,y,z)$ for the vector
$$\pmatrix{\frac{x}{C}&\frac{y}{C}&\frac{z}{C}}N\in\mathbb{Q}^...

1
vote

### On distribution of size of integer points in a subspace associated to a linear diophantine equation

This is a partial answer in one direction. Let $k=\lceil n/6\rceil$, and let
$$A=6k+1, B=10k+1, C=9k+1, D=8k+1.
$$
The numbers $A,B,C,D$ are pairwise coprime, larger than $n$, and all their ...

1
vote

### Generalized notion of divisor function?

There is an evident estimate: $f(n)= \frac{1}{2} \sum_{i=-b}^{i=b} \tau (n+i) + O(b)$ .
P.S. That is for $f(n)=d(n,\sqrt{n})$ in your initial message.

1
vote

### Occurrence of simultaneous small remainders?

This is not always true. Suppose that $c_1+c_2=c_0$. Then $pc_1+pc_2\equiv pc_0\pmod{n}$. Hence if $pc_1\bmod n$, $pc_2\bmod n$ are in $[0, n^r]$, then $pc_0\bmod n$ is in $[0, 2n^r]$, which falls ...

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