Hi,

Let $\mathbb{N}_{cop}^2$ denote the set of all pairs of coprime natural numbers. A function $f:\mathbb{C}\rightarrow\mathbb{C}$ is called asymptotically multiplicative, iff $\epsilon_{m,n}:=f(mn)-f(m)f(n)\longrightarrow 0$ uniformly as $|(n,m)|\longrightarrow\infty$ throughout $\mathbb{N}_{cop}^2$. Similarly, one could define conditioned asymptotic multiplicativity. Also, one may want to pick $f$ from some special space, e.g. $L^2$ or $C^{\infty}$ or even $\mathcal{M}$.

Now, I figured out that one may be able to study such functions by observing 'nice' matrices. In particular, let $n$ be some fixed natural number and for a matrix $A=(a_{ij})\in M_n(\mathbb{C})$ let $\tilde{a_{ij}}$ denotes the cofactor of $a_{ij}$. Is it in general possible to construct $\forall n\in\mathbb{N}$ a matrix $A\in\{M_n(\mathbb{C})}$ such that $\forall 1\leq i\leq n, 1\leq j\leq n: a_{ij}\tilde{a_{ij}}=f(i)f(j)$ ($f$ could be some arbitrary function, independent from the backgrounf I provided above)?

Thanks in advance,

efq

`$a_{ij}\widetilde{a_{ij}}=f(i)f(j)$`

: 1) Whether such factorization is possible for arbitrary $f:\mathbb{N}\rightarrow\mathbb{C}$ 2) And if not, is it possible for the case of asymptotically multiplicative functions on $\mathbb{C}$. – efq Nov 22 '09 at 8:27