MathDB

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

Problem Statement