# 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)?

Thanks in advance,

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

## 1 Answer

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
As to the link between them, I´ll try to scetch it briefly. One of the many ways to handle the property of asymptotic multiplicativity would be to observe Dirichlet-series with coefficients $f(n)$. The factorization I was asking is probably one of many ways (but a direct one) to have a matrix such that its determinant says something about the Dirichlet series. Put together, this may seem sort of unusual path, but as already said, it is just one possible path of many to explore asymptotic multiplicativity. And bringing everything to matrices may then yield pure algebraic advantage... – efq Nov 22 '09 at 17:33