Found problems: 6
Find all functions $f:\mathbb{Z}_{\ge 1}\to \mathbb{R}_{>0}$ such that for all positive integers $n$ the following relation holds: $$\sum_{d|n} f(d)^3=\left (\sum_{d|n} f(d) \right )^2,$$
where both sums are taken over the positive divisors of $n$.
[i] (Vlad Robu) [/i]
We say that a function $f : N \to N$ is increasing if $f(n) < f(m)$ whenever $n < m$, multiplicative if $f(nm) = f(n)f(m)$ whenever $n$ and $m$ are coprime, and completely multiplicative if $f(nm) = f(n)f(m)$ for all $n,m$.
(a) Prove that if $f$ is increasing then $f(n) \ge n$ for each $n$.
(b) Prove that if $f$ is increasing and completely multiplicative and $f(2) = 2$, then $f(n) = n$ for all $n$.
(c) Does (b) remain true if the word ”completely” is omitted?
Let $g : \mathbb{N} \to \mathbb{N}$ be a function satisfying:
[list]
[*] $g(xy) = g(x)g(y)$ for all $x, y \in \mathbb{N}$,
[*] $g(g(x)) = x$ for all $x \in \mathbb{N}$, and
[*] $g(x) \neq x$ for $2 \leq x \leq 2018$.
[/list]
Find the minimum possible value of $g(2)$.
Find all functions $f:\mathbb{Z}_{\ge 1}\to \mathbb{R}_{>0}$ such that for all positive integers $n$ the following relation holds: $$\sum_{d|n} f(d)^3=\left (\sum_{d|n} f(d) \right )^2,$$
where both sums are taken over the positive divisors of $n$.
[i] (Vlad Robu) [/i]
Find all functions $f : \mathbb{Q}[\sqrt{2}] \to \mathbb{Q}[\sqrt{2}]$ such that for all $x, y \in \mathbb{Q}[\sqrt{2}]$,
$$
f(xy) = f(x)f(y) \quad \text{and} \quad f(x + y) = f(x) + f(y),
$$
where $\mathbb{Q}[\sqrt{2}] = \{ a + b\sqrt{2} \mid a, b \in \mathbb{Q} \}$.
[I]Proposed by Stijn Cambie, Belgium[/i]
Consider a multiplicative function \( f \) from the positive integers to the unit disk centered at the origin, that is, \( f : \mathbb{Z}^+ \to D^2 \subseteq \mathbb{C} \) such that \( f(mn) = f(m)f(n) \). Prove that for every \( \epsilon > 0 \) and every integer \( k > 0 \), there exist \( k \) distinct positive integers \( a_1, a_2, \dots, a_k \) such that \( \text{gcd}(a_1, a_2, \dots, a_k) = k \) and \( d(f(a_i), f(a_j)) < \epsilon \) for all \( i, j = 1, \dots, k \).