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I was thinking about using the tools of functional analysis to study some subring of arithmetic functions under Dirichlet convolution. If I let $D_s$ be the ring of arithmetic functions with finite norm where $$\|f\|_{s} =\sum_{n\geq 1} \dfrac{|f(n)|}{|n^s|},$$ then this should be a Banach algebra. Now there is the Gelfand transform on this, for which I want to understand $\operatorname{Spec}(D_s)$.

I can already tell you that there should be a copy of $\{t: \Re(t)\geq \Re(s)\}\cup \{\infty\}$ should be in the Spec. Why? The following are characters: $$f\mapsto \sum_{n\geq 1} \dfrac{f(n)}{n^t} ,\quad f\mapsto f(1).$$

So my question is this: are these all the characters? Is the Gelfand transform useful in anyway to analyzing this ring?

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    $\begingroup$ I started writing an answer before I realised that I was only thinking of the case $s=0$. In that case, I think the character space of $D_0$ is given by the results mentioned in the following short paper of Hewitt and Williamson: Note on absolutely convergent Dirichlet series. Proc. Amer. Math. Soc. 8 (1957), 863--868. MathSciNet link mathscinet.ams.org/mathscinet-getitem?mr=90680 DOI doi.org/10.1090/S0002-9939-1957-0090680-X $\endgroup$
    – Yemon Choi
    Commented Nov 7, 2022 at 14:24
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    $\begingroup$ The basic point is to rephrase things in terms of the $\ell^1$-convolution algebra of the monoid $({\mathbb N},\times)$; the Gelfand representation for this algebra lives on an infinite-dimensional polydisc, whose points tell us where a given character on that monoid sends each prime. If we pick the very special characters corresponding to $n\mapsto n^{-t}$ where $t$is in the right-halfplane, and restrict the Gelfand transforms of elements in the original convolution algebra to this set of special characters, then I think we get the elements of $D_0$ (I am going from memory). $\endgroup$
    – Yemon Choi
    Commented Nov 7, 2022 at 14:31
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    $\begingroup$ @YemonChoi Oh this is a good paper! I had figured there would be some analogy with gelfand transform being Fourier transform on $\ell^1(\mathbb{Z})$, its nice to see an analogue of Wiener's theorems along these lines! I will try to read this more thoroughly and see how much of this can be salvaged for the general $D_s$ case. Thanks! $\endgroup$
    – Aareyan Manzoor
    Commented Nov 8, 2022 at 10:57

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