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

Number of distinct factors

Let me supplement the answer of i707107 for $c<1$, i.e. when we count integers with very few prime factors. Writing $$ \pi_k(n):=|\{m\in\Bbb N:\mbox{ }m\leq n,\mbox{ }\omega(m)=k\}|,$$ the ...
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

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=...
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(...
  • 8,861
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 ...
  • 90.4k
8 votes
Accepted

How many random matrices does it take to generate a matrix algebra?

At least $n^2/4$. Divide $n \times n$ matrices into four $n/2 \times n/2$ blocks, and consider the subalgebra of matrices that are zero outside the upper right block. Then because the product of any ...
  • 119k
7 votes

Number of distinct factors

Let $T(n,c)=\{m\in \mathbb{N} \ : \ m<n, \ \omega(m)<(\log\log n)^c\}$. Then we have $S(n,c)\subset T(n,c)$ and $|S(n,c)|\leq |T(n,c)|$. Also, if $T'(n,c)=\{ m\in \mathbb{N} \ : \ m<n, \ \...
  • 3,180
7 votes
Accepted

How many integers divide a number that involves just three non-zero digits?

For an integer $n$, call a prime divisor $p$ good, if the multiplicative order of $2$ modulo $p$ exceeds $p^{2/5}$, $p^2\nmid n$, and $(p-1, \varphi(n/p))<p^{1/5}$. If $p$ is a good prime divisor ...
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}\...
5 votes
Accepted

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

There are multiple books about ways to characterize the normal distribution. For instance, Bryc’s book starts with Herschel-Maxwell’s theorem: If $X$ and $Y$ are independent variables whose joint ...
  • 17.7k
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\...
  • 4,266
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 ...
  • 24.5k
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.
  • 1,644
3 votes
Accepted

Relative-totient function (2nd attempt)

I will not comment on the soundness of the approach, but I will render a subjective opinion: I don't like it. One of the reasons is that I have found most people do not have a good understanding of ...
3 votes

Relative-totient function (2nd attempt)

As I pointed out in my comment, $\Lambda(x,y) = y\phi(x)/x + O(x^{\epsilon})$. Apply this to $x_{\flat}$ instead of $x$ to get $\Lambda(x_{\flat},y) = y\phi(x_{\flat})/x_{\flat} + O(x_{\flat}^{\...
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 $...
2 votes

Exact statistics in the Frobenius coin problem

The smallest number with two representations is $ab$, so in the range you're talking about, all representations are unique. You're looking for the number of solutions to $ax+by\le m$ with $x,y\ge 0$; ...
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

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

Upper bound for an infinite series of Pochhammer Symbol

The sum is $r/(1-\alpha)^{(1+r/\alpha)}$ by the binomial theorem.
  • 14.2k
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 ...
  • 119k
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}\...
  • 8,861
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 $...
  • 779
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 $a$, $...

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