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YCor
YCor
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The Spectrumspectrum of the Banach algebra of certain arithmetic functions under Dirichlet convolution

So 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 Algebraalgebra. 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?

The Spectrum of the Banach algebra of certain arithmetic functions under Dirichlet convolution

So 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?

The spectrum of the Banach algebra of certain arithmetic functions under Dirichlet convolution

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?

Minor Math Jaxing (used $\|\cdot\|$ instead of $||\cdot||$)
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Daniele Tampieri
Daniele Tampieri
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So 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 $$||f||_{s} =\sum_{n\geq 1} \dfrac{|f(n)|}{|n^s|},$$$$\|f\|_{s} =\sum_{n\geq 1} \dfrac{|f(n)|}{|n^s|},$$ 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?

So 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?

So 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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Aareyan Manzoor
Aareyan Manzoor
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The Spectrum of the Banach algebra of certain arithmetic functions under Dirichlet convolution

So 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?