1
$\begingroup$

A function is called quasi-multiplicative by many authors if $f(m)f(n)=f(1)f(mn)$, a slight generalization of multiplicativity. (Basically a multiplicative function times a constant is a quasi-multiplicative function.)

Lahiri [1] introduces the notion of a hypo-multiplicative function as one where $$ \left|\begin{matrix} f(u_1v_1) & \ldots & f(u_1v_n) \\ \vdots & \ddots & \vdots \\ f(u_nv_1) & \ldots & f(u_nv_n) \end{matrix}\right|=0 $$ for some $n$ with any $u_i,v_i$ subject to $\gcd(u_i,v_j)=1$ for all $1\le i,j\le n.$ This is a generalization of being quasi-multiplicative, which is the 2×2 case; in fact, any linear combination of quasi-multiplicative functions is hypo-multiplicative.

Lahiri mentions that he does not know if the condition is a biimplication. I was wondering if this was now known, or perhaps obvious to someone here. Are there hypo-multiplicative functions which are not linear combinations of quasi-multiplicative functions?


This question was originally posted to math.stackexchange.com but I thought it would get a better response here.

[1] D. B. Lahiri, "Hypo-multiplicative number-theoretic functions", Aequationes Mathematicae 9:2-3 (1968), pp. 184–192. MR0337736

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.