All Questions
12 questions
1
vote
0
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150
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What are alternative mathematical definitions of observers beyond Bennett and Hoffman's framework?
Motivation:
This question is inspired by a talk from Avi Wigderson given on Randomness, where the idea that the randomness is in the eye of the observer is suggested.
In the study of information ...
- 3,064
7
votes
0
answers
222
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Projected polar chessboard measure convergence in total variation?
$\newcommand\R{\mathbb R}\newcommand\C{\mathbb C}\newcommand\ga{\gamma}$For natural $n$, let $E_n$ be the set of all points in $\R^2$ with "polar coordinates" $(r,t)$ in the set
$$F_n:=\...
- 128k
32
votes
3
answers
12k
views
What is the Katz-Sarnak philosophy?
It has been recently mentioned by a speaker (his talk is completely not relevant to random matrix theory/RMT though) that modern statistics, especially random matrices theory, will help solving some ...
- 8,071
17
votes
13
answers
6k
views
Probability in number theory
I am hearing that there are some great applications of probability theory (or more general measure theory) to number theory. Could anyone recommend some good book(s) on that (or other types of ...
7
votes
1
answer
465
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A theorem by Harald Cramér?
In the paper “On the order of magnitude of the difference between consecutive prime numbers” by Harald Cramér there is the following statement:
Suppose $\{X_n\}_{n=2}^\infty$ is a sequence of ...
- 2,618
3
votes
3
answers
330
views
Reference request: probability that d numbers are coprime
The following theorem can be found in Hardy-Wright (Theorem 459), except that they state it only for $d=2$. Do you know of a reference where the proof of this general statement is written?
Theorem: ...
- 402
8
votes
2
answers
512
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The average of reciprocal binomials
This question is motivated by the MO problem here. Perhaps it is not that difficult.
Question. Here is an cute formula.
$$\frac1n\sum_{k=0}^{n-1}\frac1{\binom{n-1}k}=\sum_{k=1}^n\frac1{k2^{n-k}}...
- 43.2k
3
votes
2
answers
411
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When are "normal" functions normal?
I expected that the fractional part of f(n), n being an integer, would be distributed uniformly over [0,1] (for positive functions - otherwise take [-1,1]) for any run-of-the-mill function, except ...
- 4,829
8
votes
2
answers
537
views
Famous results about the value of a given limit assuming it exists
Chebyshev got famous showing that if the limit $l:=\lim_{x\to\infty}\frac{\pi(x)}{x/\log x}$ exists, then necessarily $l=1$, constituting a major breakthrough towards a proof of the famous prime ...
- 6,984
7
votes
2
answers
1k
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What is a random number? (poll experiment) [closed]
Imagine the following experiment: you wait say at a subway exit, and ask everyone passing "please tell me a number" (positive integer, of course). You do this day after day, until you reach say 1M ...
- 1,363
6
votes
3
answers
938
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Uniformly distributed sequence in $\mathbb{R}$
We say that a sequence $(x_n)_{n=1}^\infty \subseteq \mathbb{R}$ is "uniformly distributed in $[a,b]$", with $a < b$, if $(x_n)_{n=1}^\infty \cap [a,b] \neq \varnothing$ and
$$\lim_{N \to \infty} \...
- 61
11
votes
0
answers
282
views
Reference request: a combinatoric result [closed]
When I tried to construct a counterexample in my research, I encountered the following result, which should be true.
Let $m=m(n)$ be a function that grows faster than $\sqrt n$, so $m(n) = \omega(\...
- 211