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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).

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