# Tagged Questions

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

Define $f(n) = |\{m : m\le n, \exists k \text{ s.t. }\phi(k) = m\}|$. Clearly, $f(n)\le \left\lfloor \frac{n}{2}\right\rfloor + 1$ since $\phi(n)$ is even for all $n > 2$. Is ...
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### The lonely molecule

Suppose $n$ air molecules (infinitesimal points) are bouncing around in a unit $d$-dimensional cube, with perfectly elastic wall collisions. Let $k=n^{\frac{1}{d}}$. For example, in 3D, $d=3$, with ...
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### Does the set of automorphisms of a cyclic group exhibit some sense of randomness?

I prefer to proceed with a concrete example if I may. I appreciate that the answer might well be better explained with group theory, geometry and/or notions from probability theory, which I welcome. ...
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### Not-lonely runners

The lonely runner conjecture has several formulations. They all involve a number $n$ runners running on a circular track, each with a different speeds, and the conjecture is that each runner is ...
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### Mellin transform of time-shifted function

The Mellin transform of a function $f(x)$ can be written as $$\mathcal M[f(x);z]=\int_0^\infty f(x)x^{z-1} dx$$ Is there a simple expression for the Mellin transform of the function $f(x-x_0)$? ...
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### Analogy between Integers and Permutations

I am reading Andrew Granville's Anatomy of Integers and Permutations where it is argued the factorization of a permutation into disjoint cycles is analogous to the factorization of a number into prime ...
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### Random pseudoprimes vs. primes

(Edit. What I called "pseudoprimes" are known as "CramÃ©r random primes" in the literature, of which I was unaware.) Say that a set $S$ of natural numbers is a set of pseudoprimes if they are (a) ...
Hello, I would like to know whether there is a way, thanks to the prime number theorem, to give some kind of an equivalent of the probability that two positive integers $p$ and $q$ less than a given ...