# Multiplicative quasi-characters on ℚ

I want references on multiplicative quasi-characters on $$\mathbb{Q}$$, especially having growth on natural numbers bounded with $$|\chi(m)| \le m$$. Results in number theory? Use in visualization of data, of numbers themselves, etc.? Some people who do any research related to such quasi-characters? Any information of this sort is appreciated.

Quasi-characters (sometimes referred to as simply to characters) are group homomorphisms from $$\mathbb{Q}^\times$$ to $$\mathbb{C}^\times$$ or, in this context, such complex-valued functions $$\chi$$ on natural numbers that $$\chi(1) = 1$$ and $$\chi(mn) = \chi(m)\chi(n)$$ for all $$m$$ and $$n$$. For $$\chi$$ on positive numbers, it ’s sufficient to specify $$\chi(p)$$ for all primes only.

The only thing of value I can see is an oblique connection from André Weil’s Basic Number Theory Lemma 3 (in §1.3) that states that for each such function on natural numbers at least one of the three statements is true:

• $$|\chi(m)| \le 1$$ for all $$m$$;
• There exists such $$\lambda > 0$$ that $$|\chi(m)| = m^\lambda$$ for all $$m$$;
• There is no such constant $$A$$ that $$|\chi(m+n)| \le A\max(|\chi(m)|, |\chi(n)|)$$ for all $$m$$ and $$n$$.

I am especially interested in the last case – no [weak] subadditivity (albeit growth is restricted).