# Asymptotically multiplicative functions and matrices

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)?

efq

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There is something wrong with that post. The preview doesn´t have any issues! –  efq Nov 22 '09 at 4:55
OK, I think I fixed it. It was the underscore bug. –  efq Nov 22 '09 at 5:01
What is "script M" meant to be? What measure are you using for $L^2$? –  Yemon Choi Nov 22 '09 at 7:07
$\mathcal{M}$ is meant to stand for meromorphic. I didn´t specify it, nor the measure in $L^2$, because I meant these as optional conditions on those functions $f$ that might turn out to be of interest at some point. I´m still at the stage of exploring options. What is here important, is the property of asymptotic multiplicativity and the factorization $a_{ij}\widetilde{a_{ij}}=f(i)f(j)$: 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
It may very well turn out, that such factorization is not possible for every $n\in\mathbb{N}$ at all! –  efq Nov 22 '09 at 8:28

I don't understand several points in your question. Firstly, your function $f$ is only defined on the set of natural numbers, so talking about it belonging to $L^2$ or $C^\infty$ or ${\mathcal M}$ seems misleading.

Bear in mind that any function defined on the natural numbers can be extended to a smooth function defined on ${\mathbb C}$.

Secondly, I don't understand the quantifiers in your second question. Are you specifying a function $f$, a number $n$, and then asking for an $n\times n$ matrix A which has the properties you require? Or are you asking which matrices $A$ have the property that $a_{ij} \widetilde{a_{ij}}$ factorizes as $f(i)f(j)$ for some function $f$? The identity matrix ought to be such an example, but presumably you want others.

Could you say a little more about why the second part should be relevant to your notion of "asymptotically multiplicative"?

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First of all, $f$ is defined on $\mathbb{C}$ and admits the property of asymptotic multiplicativity on the subset $mathcal{N}_{cop}^2\subset\mathbb{C\times C}$ The idea is to study the family of all such functions. However, this subset may turn out to be quite large, so I gave rather as examples some additional properties one may wish to require from those functions. I don´t see why this is misleading. –  efq Nov 22 '09 at 8:58
Regarding the factorization, I added $\forall n\in\mathbb{N}$ to my post for more clarity. Of course, my $f$ is given. So, I want to find a matrix $A\in M_n(\mathbb{C})$ for each $n\in\mathbb{N}$ so that the the factorization given above is fulfilled. –  efq Nov 22 '09 at 8:59
This factorization may even work for arbitrary $f:\matbb{N}\rightarrow\mathbb{C}$. Or it may work only for some class of functions, to which my $f$ does or does not belong. Or it may not work at all. Note that the condition $\forall n\in \mathbb{N}$ is essential. –  efq Nov 22 '09 at 9:03
Perhaps I was hasty in saying "misleading". My point remains, that one could imagine having very different functions defined on $\mathbb C$, which coincide on the set ${\mathbb N}$. Now if you demand that your functions are, say, meromorphic, then there will be some restrictions. But you also said something about $C^\infty$ or $L^2$, and in those classes knowing the values of the function on the positive integers says almost nothing useful about the function in general. –  Yemon Choi Nov 22 '09 at 12:44
$L^2$, $\mathcal{M}$, $\C^{\infty}$ were rather for sketchy purposes,and probably you are right that it was somewhat misleading in this way. I agree that the class of asymptotically multiplicative functions seems to be quite wide and requiring meromorphic could be a restriction of interest. I haven´t really thought about the additional restrictions yet. Also, I guess I should have stated those problems separately. –  efq Nov 22 '09 at 17:24