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It is helpful to state things first in terms of abstract harmonic analysis on locally compact abelian (=LCA) groups might make things clearer.

For any LCA group $A$ let $\widehat{A}$ be the group of it's unitary characters. If $f$ is a function on $A$ it's Fourier transform is $\chi \mapsto \widehat{f}(\chi) = \int_{A} f(x)\chi^{-1}(x)dx$ with $dx$ a fixed Haar measure on $A$. My Pontryagin duality is a unique measure $d\chi$ on the LCA group $\widehat{A}$ dual to $dx$ satisfying the Fourier inversion

$$ f(e) = \int_{\widehat{A}} \widehat{f}(\chi) d\chi $$

$$ f(y) = \int_{\widehat{A}} \widehat{f}(\chi)\chi(y) d\chi $$

$\mathbb{Q}_p^{\times}$ is your abelian group, under multiplication. It's Pontryagin dual is the space of unitary complex characters $ \widehat{\mathbb{Q}_p} =\{ \chi: \mathbb{Q}_p^{\times} \to \mathbb{T} \}$ with $\mathbb{T}$ the complex circle. For any smooth compactly supported function on $\mathbb{Q}_p^{\times}$ it's abstract Fourier transform is the Mellin transform $\widehat{f}(\chi) = \int_{\mathbb{Q}_p^{\times}} f(x) \chi(x) dx$

Because $\mathbb{Q}_p^{\times} = \mathbb{Z}_p^{\times} \times p^{\mathbb{Z}}$ it's unitary dual is $\widehat{\mathbb{Z}_p^{\times}} \times \widehat{\mathbb{Z}}$. The unramified characters are the characters which are trivial on $\mathbb{Z}_p^{\times}$, i.e. they are restrictions to $\{ 1 \} \times \widehat{\mathbb{Z}}$. The unitary characters of $\mathbb{Z}$ can be identified with unit norm complex numbers $\mathbb{T} = \{ |z|=1 \}$. The complex parameter refers to the exponent $s$ in the unitary character determined by $ | \cdot |^{s}:p \mapsto |p|_{p}^s=p^{-s}$ (or however you want to normalize it, but this is natural for number theory). To conform with Ngo's noration let us make a change of variables $t = p^{-s}$ and call this unitary character $\chi_t$. So we should think of the function "$\frac{1}{1-p^{-s}}$" more precisely as a function $L$ on the space of unitary characters:

$$\chi_t \mapsto L(\chi_t) = \frac{1}{1-\chi_t(p)}$$ if $\chi_t$ is an unramified chatacter $$ \eta \mapsto L(\eta) = 0 $$ If $\eta$ is not unramified.

The dual measure $d\chi_{t}$ is always a Haar measure on the unitary dual, which in our case is $\mathbb{T}$, so is a multiple of $dt/t$. Explicitly it looks like $\frac{\mathrm{ln}(p)}{2\pi i} \frac{dt}{t}$ (because the contour is over a parameter between $\frac{-\pi}{\mathrm{ln}(p)} $ and $\frac{\pi}{\mathrm{ln}(p)}$).

We can write $L(\chi_t) = \chi_{t}(e) + \chi_{t}(p) + \chi_{t}(p^2) + \cdots = \chi_{t}(e) + l_p \chi_{t}(e) + l_{p^2} \chi_{t}(e) + \cdots$ where $l_p$ is translation by $p$. Strictly speaking this is not a polynomial function on the space of unitary characters but a rational one, but formally we can do everything for the terms in the sum and then take a limit.

Thus it is enough to Mellin-invert the function $\widehat{f_0}: \chi_t \mapsto \chi_{t}(e) = 1$ and deduce the rest of it by translating. Using the Fourier inversion formula and the fact that $\widehat{f_0}$ is supported on $\mathbb{Z}_p^{\times}$-invariant characters implies that $f_0$ is $\mathbb{Z}_p^{\times}$-invariant. Thus it is enough to know it's value on $\mathbb{Z}_p^{\times}$ and it's translates $p^n\mathbb{Z}_p^{\times}$.

$$f_0(e)=\mathrm{const} \int_{ |t|=1 } \chi_t(e) \frac{dt}{t} = \mathrm{const} \int_{ |t|=1 } 1 \frac{dt}{t}$$

Thus $f_0(\mathbb{Z}_p^{\times})=1$. Arguing by translation by either $p$ or $p^{-1}$ we obtain

$$f_0(p) = \mathrm{const} \int_{ |t|=1 } \chi_t(p)\frac{dt}{t} = \int_{ |t|=1 } dt=0$$ or

$$f(p^{-1}=\mathrm{const} \int_{ |t|=1 } \chi_t(p^{-1})\frac{dt}{t} = \int_{ |t|=1 } t^{-2}dt =0.$$

So $f_0 = 1_{ \mathbb{Z}_p^{\times}}$.

Arguing similarly with $\chi_{t} \mapsto l_p(\chi)(e)$ produces a function $f_{1}$ such that $l_p(f)(\mathbb{Z}_p^{\times}) =1$ and is invariant under and supported in $\mathbb{Z}_p^{\times}$. In other words it is the characteristic function of the coset $p \mathbb{Z}_p^{\times}$. And so on for the rest of the terms. Adding them all up yields $ \sum_n f_n = 1_{\mathbb{Z}_p - \{ 0 \}}$ as exepcted.

Edit:

The inversion of $F(s)$, in your notation is

$$f(x) = \frac{\mathrm{ln}(p)}{2 \pi i} \int_{ \{ s=ir \ : r \in (-\pi/\mathrm{ln}(p), \pi/\mathrm{ln}(p)] \}} F(s)|x|^{-s} ds = \frac{\mathrm{ln}(p)}{2 \pi i} \int_{ \{ s=ir \ : r \in (-\pi/\mathrm{ln}(p), \pi/\mathrm{ln}(p)] \}} F(s)p^{\mathrm{val}(x)s} ds$$

Here if $x=p^nu$ where $u \in \mathbb{Z}_p^{\times}$ is a unit $\mathrm{val}(x)=n$. You can evaluate this integral in the normal way, you don't ignore poles. It seems like the closest answer you got was the constant $\frac{1}{\mathrm{ln}(p)}$, but this is incomplete because $f(x)$ is not supported on $\mathbb{Q}_p^{\times}$ but rather on $\{ \mathrm{val}(x) \geq 0 \} $. You can see this because in the series expansion of the rational function $\frac{1}{1-p^{-s}} = 1 + p^{-s}+p^{-2s} + \cdots $ the terms consist of polynomials in the variable $p^{-s}$ and each integral $\int_{ \{ |s|=1 \} } p^{-ns}|x|^{-s}$ is non-zero only when the valuation of $x$ is $n$ for $n \geq 0$. Finally the $\mathrm{ln}(p)$ terms just comes to account for the fact that we are parametrizing the complex circle by $r \mapsto p^{-ir}$ so as to give complex circle measure $1$.

=================

It is helpful to state things first in terms of abstract harmonic analysis on locally compact abelian (=LCA) groups might make things clearer.

For any LCA group $A$ let $\widehat{A}$ be the group of it's unitary characters. If $f$ is a function on $A$ it's Fourier transform is $\chi \mapsto \widehat{f}(\chi) = \int_{A} f(x)\chi^{-1}(x)dx$ with $dx$ a fixed Haar measure on $A$. My Pontryagin duality is a unique measure $d\chi$ on the LCA group $\widehat{A}$ dual to $dx$ satisfying the Fourier inversion

$$ f(e) = \int_{\widehat{A}} \widehat{f}(\chi) d\chi $$

$$ f(y) = \int_{\widehat{A}} \widehat{f}(\chi)\chi(y) d\chi $$

$\mathbb{Q}_p^{\times}$ is your abelian group, under multiplication. It's Pontryagin dual is the space of unitary complex characters $ \widehat{\mathbb{Q}_p} =\{ \chi: \mathbb{Q}_p^{\times} \to \mathbb{T} \}$ with $\mathbb{T}$ the complex circle. For any smooth compactly supported function on $\mathbb{Q}_p^{\times}$ it's abstract Fourier transform is the Mellin transform $\widehat{f}(\chi) = \int_{\mathbb{Q}_p^{\times}} f(x) \chi(x) dx$

Because $\mathbb{Q}_p^{\times} = \mathbb{Z}_p^{\times} \times p^{\mathbb{Z}}$ it's unitary dual is $\widehat{\mathbb{Z}_p^{\times}} \times \widehat{\mathbb{Z}}$. The unramified characters are the characters which are trivial on $\mathbb{Z}_p^{\times}$, i.e. they are restrictions to $\{ 1 \} \times \widehat{\mathbb{Z}}$. The unitary characters of $\mathbb{Z}$ can be identified with unit norm complex numbers $\mathbb{T} = \{ |z|=1 \}$. The complex parameter refers to the exponent $s$ in the unitary character determined by $ | \cdot |^{s}:p \mapsto |p|_{p}^s=p^{-s}$ (or however you want to normalize it, but this is natural for number theory). To conform with Ngo's noration let us make a change of variables $t = p^{-s}$ and call this unitary character $\chi_t$. So we should think of the function "$\frac{1}{1-p^{-s}}$" more precisely as a function $L$ on the space of unitary characters:

$$\chi_t \mapsto L(\chi_t) = \frac{1}{1-\chi_t(p)}$$ if $\chi_t$ is an unramified chatacter $$ \eta \mapsto L(\eta) = 0 $$ If $\eta$ is not unramified.

The dual measure $d\chi_{t}$ is always a Haar measure on the unitary dual, which in our case is $\mathbb{T}$, so is a multiple of $dt/t$. Explicitly it looks like $\frac{\mathrm{ln}(p)}{2\pi i} \frac{dt}{t}$ (because the contour is over a parameter between $\frac{-\pi}{\mathrm{ln}(p)} $ and $\frac{\pi}{\mathrm{ln}(p)}$).

We can write $L(\chi_t) = \chi_{t}(e) + \chi_{t}(p) + \chi_{t}(p^2) + \cdots = \chi_{t}(e) + l_p \chi_{t}(e) + l_{p^2} \chi_{t}(e) + \cdots$ where $l_p$ is translation by $p$. Strictly speaking this is not a polynomial function on the space of unitary characters but a rational one, but formally we can do everything for the terms in the sum and then take a limit.

Thus it is enough to Mellin-invert the function $\widehat{f_0}: \chi_t \mapsto \chi_{t}(e) = 1$ and deduce the rest of it by translating. Using the Fourier inversion formula and the fact that $\widehat{f_0}$ is supported on $\mathbb{Z}_p^{\times}$-invariant characters implies that $f_0$ is $\mathbb{Z}_p^{\times}$-invariant. Thus it is enough to know it's value on $\mathbb{Z}_p^{\times}$ and it's translates $p^n\mathbb{Z}_p^{\times}$.

$$f_0(e)=\mathrm{const} \int_{ |t|=1 } \chi_t(e) \frac{dt}{t} = \mathrm{const} \int_{ |t|=1 } 1 \frac{dt}{t}$$

Thus $f_0(\mathbb{Z}_p^{\times})=1$. Arguing by translation by either $p$ or $p^{-1}$ we obtain

$$f_0(p) = \mathrm{const} \int_{ |t|=1 } \chi_t(p)\frac{dt}{t} = \int_{ |t|=1 } dt=0$$ or

$$f(p^{-1}=\mathrm{const} \int_{ |t|=1 } \chi_t(p^{-1})\frac{dt}{t} = \int_{ |t|=1 } t^{-2}dt =0.$$

So $f_0 = 1_{ \mathbb{Z}_p^{\times}}$.

Arguing similarly with $\chi_{t} \mapsto l_p(\chi)(e)$ produces a function $f_{1}$ such that $l_p(f)(\mathbb{Z}_p^{\times}) =1$ and is invariant under and supported in $\mathbb{Z}_p^{\times}$. In other words it is the characteristic function of the coset $p \mathbb{Z}_p^{\times}$. And so on for the rest of the terms. Adding them all up yields $ \sum_n f_n = 1_{\mathbb{Z}_p - \{ 0 \}}$ as exepcted.

Although you say you're not interested in the algebraic aspects of things, restating things in terms of abstract harmonic analysis on locally compact abelian (=lca) groups might make things clearer.

For any locally compact abelian group $A$ and $\widehat{A}$ the group of it's unitary characters. $\chi \mapsto \widehat{f}(\chi) = \int_{A} f(x)\chi(x)dx$ with $dx$ a fixed haar measure on $A$. there is a measure $d\chi$ on the lca $\widehat{A}$ dual to $dx$ satisfying

$$ f(e) = \int_{\widehat{A}} \widehat{f}(\chi) d\chi $$

$\mathbb{Q}_p^{\times}$ is your abelian group, under multiplication. It's Pontryagin dual is the space of unitary complex characters $ \widehat{\mathbb{Q}_p} =\{ \chi: \mathbb{Q}_p^{\times} \to \mathbb{T} \}$ with $\mathbb{T}$ the complex circle. For any smooth compactly supported function on $\mathbb{Q}_p^{\times}$ it's abstract Fourier transform is the function $\widehat{f}(\chi) = \int_{\mathbb{Q}_p^{\times}} f(x) \chi(x) dx$

Because $\mathbb{Q}_p^{\times} = \mathbb{Z}_p^{\times} \times p^{\mathbb{Z}}$ it's unitary dual is $\widehat{\mathbb{Z}_p^{\times}} \times \widehat{\mathbb{Z}}$. The unitary dual of $\mathbb{Z}$ is $\mathbb{T}$. The complex parameter refers to the unitary character determined by $p \mapsto |p|_{p}^s=p^{-s}$ (or however you want to normalize it, but this is natural for number theory). Call this unitary character $\chi_s$. The remark about the unramified characters means it is a characters of $\mathbb{Q}_p^{\times}$ that is invariant under $\mathbb{Z}_p^{\times}$. Therefore you may think of the function "$\frac{1}{1-p^{-s}}$" more explicitly as a function $L$ on the unramified unitary dual

$$\chi_s \mapsto L(\chi_s) = \frac{1}{1-\chi_s(p)}$$ if $\chi_s$ unramified. $$ \eta \mapsto L(\eta) = 0 $$ $\eta$ not unramified.

The dual measure is always a haar measure on $\mathbb{T}$, which in this case looks like $\frac{\mathrm{ln}(p)}{2\pi i} \frac{ds}{s}$ (because the contour is over a parameter between $\frac{-\pi}{\mathrm{ln}(p)} $ and $\frac{\pi}{\mathrm{ln}(p)}$.

We can write $L(\chi_s) = 1 + \chi_{s}(p) + \chi_{s}(p^2) + \cdots = \chi{s}(e) + l_p \chi_{s}(e) + l_{p^2} \chi_{s}(e) + \cdots$ where $l_p$ is translation by $p$

We may think of the function $\chi_s \mapsto \chi_{s}(e) = 1$ as a function on the set of characters restricted to $\mathbb{Z}_p^{\times}$. Inverting this, gives a function $ f_{0}(e) = \frac{\mathrm{ln}(p)}{2\pi i} \int_{\mathbb{T}} 1 \frac{ds}{s}$ just gives you the normalized measure of the circle which is $1$. Because $\chi_s(\mathbb{Z}_p^{\times})=1$, $f_0$ must be invariant under, and supported in $\mathbb{Z}_p^{\times}$

Similarly inverting $\chi \mapsto l_p(\chi)(e)$ produces function $f_{1}$ such that $l_p(f)(\mathbb{Z}_p^{\times}) =1$ and is invariant under $\mathbb{Z}_p^{\times}$. In other words, this is the characteristic function of the coset $p \mathbb{Z}_p^{\times}$. Thus $ \sum_n f_n = 1_{\mathbb{Z}_p}$

It is helpful to state things first in terms of abstract harmonic analysis on locally compact abelian (=LCA) groups might make things clearer.

For any LCA group $A$ let $\widehat{A}$ be the group of it's unitary characters. If $f$ is a function on $A$ it's Fourier transform is $\chi \mapsto \widehat{f}(\chi) = \int_{A} f(x)\chi^{-1}(x)dx$ with $dx$ a fixed Haar measure on $A$. My Pontryagin duality is a unique measure $d\chi$ on the LCA group $\widehat{A}$ dual to $dx$ satisfying the Fourier inversion

$$ f(e) = \int_{\widehat{A}} \widehat{f}(\chi) d\chi $$

$$ f(y) = \int_{\widehat{A}} \widehat{f}(\chi)\chi(y) d\chi $$

$\mathbb{Q}_p^{\times}$ is your abelian group, under multiplication. It's Pontryagin dual is the space of unitary complex characters $ \widehat{\mathbb{Q}_p} =\{ \chi: \mathbb{Q}_p^{\times} \to \mathbb{T} \}$ with $\mathbb{T}$ the complex circle. For any smooth compactly supported function on $\mathbb{Q}_p^{\times}$ it's abstract Fourier transform is the Mellin transform $\widehat{f}(\chi) = \int_{\mathbb{Q}_p^{\times}} f(x) \chi(x) dx$

Because $\mathbb{Q}_p^{\times} = \mathbb{Z}_p^{\times} \times p^{\mathbb{Z}}$ it's unitary dual is $\widehat{\mathbb{Z}_p^{\times}} \times \widehat{\mathbb{Z}}$. The unramified characters are the characters which are trivial on $\mathbb{Z}_p^{\times}$, i.e. they are restrictions to $\{ 1 \} \times \widehat{\mathbb{Z}}$. The unitary characters of $\mathbb{Z}$ can be identified with unit norm complex numbers $\mathbb{T} = \{ |z|=1 \}$. The complex parameter refers to the exponent $s$ in the unitary character determined by $ | \cdot |^{s}:p \mapsto |p|_{p}^s=p^{-s}$ (or however you want to normalize it, but this is natural for number theory). To conform with Ngo's noration let us make a change of variables $t = p^{-s}$ and call this unitary character $\chi_t$. So we should think of the function "$\frac{1}{1-p^{-s}}$" more precisely as a function $L$ on the space of unitary characters:

$$\chi_t \mapsto L(\chi_t) = \frac{1}{1-\chi_t(p)}$$ if $\chi_t$ is an unramified chatacter $$ \eta \mapsto L(\eta) = 0 $$ If $\eta$ is not unramified.

The dual measure $d\chi_{t}$ is always a Haar measure on the unitary dual, which in our case is $\mathbb{T}$, so is a multiple of $dt/t$. Explicitly it looks like $\frac{\mathrm{ln}(p)}{2\pi i} \frac{dt}{t}$ (because the contour is over a parameter between $\frac{-\pi}{\mathrm{ln}(p)} $ and $\frac{\pi}{\mathrm{ln}(p)}$).

We can write $L(\chi_t) = \chi_{t}(e) + \chi_{t}(p) + \chi_{t}(p^2) + \cdots = \chi_{t}(e) + l_p \chi_{t}(e) + l_{p^2} \chi_{t}(e) + \cdots$ where $l_p$ is translation by $p$. Strictly speaking this is not a polynomial function on the space of unitary characters but a rational one, but formally we can do everything for the terms in the sum and then take a limit.

Thus it is enough to Mellin-invert the function $\widehat{f_0}: \chi_t \mapsto \chi_{t}(e) = 1$ and deduce the rest of it by translating. Using the Fourier inversion formula and the fact that $\widehat{f_0}$ is supported on $\mathbb{Z}_p^{\times}$-invariant characters implies that $f_0$ is $\mathbb{Z}_p^{\times}$-invariant. Thus it is enough to know it's value on $\mathbb{Z}_p^{\times}$ and it's translates $p^n\mathbb{Z}_p^{\times}$.

$$f_0(e)=\mathrm{const} \int_{ |t|=1 } \chi_t(e) \frac{dt}{t} = \mathrm{const} \int_{ |t|=1 } 1 \frac{dt}{t}$$

Thus $f_0(\mathbb{Z}_p^{\times})=1$. Arguing by translation by either $p$ or $p^{-1}$ we obtain

$$f_0(p) = \mathrm{const} \int_{ |t|=1 } \chi_t(p)\frac{dt}{t} = \int_{ |t|=1 } dt=0$$ or

$$f(p^{-1}=\mathrm{const} \int_{ |t|=1 } \chi_t(p^{-1})\frac{dt}{t} = \int_{ |t|=1 } t^{-2}dt =0.$$

So $f_0 = 1_{ \mathbb{Z}_p^{\times}}$.

Arguing similarly with $\chi_{t} \mapsto l_p(\chi)(e)$ produces a function $f_{1}$ such that $l_p(f)(\mathbb{Z}_p^{\times}) =1$ and is invariant under and supported in $\mathbb{Z}_p^{\times}$. In other words it is the characteristic function of the coset $p \mathbb{Z}_p^{\times}$. And so on for the rest of the terms. Adding them all up yields $ \sum_n f_n = 1_{\mathbb{Z}_p - \{ 0 \}}$ as exepcted.

Although you say you're not interested in the algebraic aspects of things, restating things in terms of abstract harmonic analysis on locally compact abelian (=lca) groups might make things clearer.

For any locally compact abelian group $A$ and $\widehat{A}$ the group of it's unitary characters. $\chi \mapsto \widehat{f}(\chi) = \int_{A} f(x)\chi(x)dx$ with $dx$ a fixed haar measure on $A$. there is a measure $d\chi$ on the lca $\widehat{A}$ dual to $dx$ satisfying

$$ f(e) = \int_{\widehat{A}} \widehat{f}(\chi) d\chi $$

$\mathbb{Q}_p^{\times}$ is your abelian group, under multiplication. It's Pontryagin dual is the space of unitary complex characters $ \widehat{\mathbb{Q}_p} =\{ \chi: \mathbb{Q}_p^{\times} \to \mathbb{T} \}$ with $\mathbb{T}$ the complex circle. For any smooth compactly supported function on $\mathbb{Q}_p^{\times}$ it's abstract Fourier transform is the function $\widehat{f}(\chi) = \int_{\mathbb{Q}_p^{\times}} f(x) \chi(x) dx$

Because $\mathbb{Q}_p^{\times} = \mathbb{Z}_p^{\times} \times p^{\mathbb{Z}}$ it's unitary dual is $\widehat{\mathbb{Z}_p^{\times}} \times \widehat{\mathbb{Z}}$. The unitary dual of $\mathbb{Z}$ is $\mathbb{T}$. The complex parameter refers to the unitary character determined by $p \mapsto |p|_{p}^s=p^{-s}$ (or however you want to normalize it, but this is natural for number theory). Call this unitary character $\chi_s$. The remark about the unramified characters means it is a characters of $\mathbb{Q}_p^{\times}$ that is invariant under $\mathbb{Z}_p^{\times}$. Therefore you may think of the function "$\frac{1}{1-p^{-s}}$" more explicitly as a function $L$ on the unramified unitary dual

$$\chi_s \mapsto L(\chi_s) = \frac{1}{1-\chi_s(p)}$$ if $\chi_s$ unramified. $$ \eta \mapsto L(\eta) = 0 $$ $\eta$ not unramified.

The dual measure is always a haar measure on $\mathbb{T}$, which in this case looks like $\frac{\mathrm{ln}(p)}{2\pi i} \frac{ds}{s}$ (because the contour is over a parameter between $\frac{-\pi}{\mathrm{ln}(p)} $ and $\frac{\pi}{\mathrm{ln}(p)}$.

We can write $L(\chi_s) = 1 + \chi_{s}(p) + \chi_{s}(p^2) + \cdots = \chi{s}(e) + l_p \chi_{s}(e) + l_{p^2} \chi_{s}(e) + \cdots$ where $l_x$ is translation by $x$. Or in other words we write the sum with terms $\chi \mapsto \chi(p^n)$ considered as a function supported on $ \{ p^n \}$.

We may think of the function $\chi \mapsto \chi_{s}(e) = 1$ as a function on $p^{\mathbb{Z}}$ supported at $\{ p^{0} \}$. Inverting this, gives a function $ f_{0}(e) = \frac{\mathrm{ln}(p)}{2\pi i} \int_{\mathbb{T}} 1 \frac{ds}{s}$ just gives you the normalized measure of the circle which is $1$. Because $\chi_s(\mathbb{Z}_p^{\times})=1$, $f_0$ must be invariant under, and supported in $\mathbb{Z}_p^{\times}$

Similarly inverting $\chi \mapsto l_p(\chi)(e)$ produces function $f_{1}$ such that $l_p(f)(\mathbb{Z}_p^{\times}) =1$ and is invariant under $\mathbb{Z}_p^{\times}$. In other words, this is the characteristic function of the coset $p \mathbb{Z}_p^{\times}$. Thus $ \sum_n f_n = 1_{\mathbb{Z}_p}$

Although you say you're not interested in the algebraic aspects of things, restating things in terms of abstract harmonic analysis on locally compact abelian (=lca) groups might make things clearer.

For any locally compact abelian group $A$ and $\widehat{A}$ the group of it's unitary characters. $\chi \mapsto \widehat{f}(\chi) = \int_{A} f(x)\chi(x)dx$ with $dx$ a fixed haar measure on $A$. there is a measure $d\chi$ on the lca $\widehat{A}$ dual to $dx$ satisfying

$$ f(e) = \int_{\widehat{A}} \widehat{f}(\chi) d\chi $$

$\mathbb{Q}_p^{\times}$ is your abelian group, under multiplication. It's Pontryagin dual is the space of unitary complex characters $ \widehat{\mathbb{Q}_p} =\{ \chi: \mathbb{Q}_p^{\times} \to \mathbb{T} \}$ with $\mathbb{T}$ the complex circle. For any smooth compactly supported function on $\mathbb{Q}_p^{\times}$ it's abstract Fourier transform is the function $\widehat{f}(\chi) = \int_{\mathbb{Q}_p^{\times}} f(x) \chi(x) dx$

Because $\mathbb{Q}_p^{\times} = \mathbb{Z}_p^{\times} \times p^{\mathbb{Z}}$ it's unitary dual is $\widehat{\mathbb{Z}_p^{\times}} \times \widehat{\mathbb{Z}}$. The unitary dual of $\mathbb{Z}$ is $\mathbb{T}$. The complex parameter refers to the unitary character determined by $p \mapsto |p|_{p}^s=p^{-s}$ (or however you want to normalize it, but this is natural for number theory). Call this unitary character $\chi_s$. The remark about the unramified characters means it is a characters of $\mathbb{Q}_p^{\times}$ that is invariant under $\mathbb{Z}_p^{\times}$. Therefore you may think of the function "$\frac{1}{1-p^{-s}}$" more explicitly as a function $L$ on the unramified unitary dual

$$\chi_s \mapsto L(\chi_s) = \frac{1}{1-\chi_s(p)}$$ if $\chi_s$ unramified. $$ \eta \mapsto L(\eta) = 0 $$ $\eta$ not unramified.

The dual measure is always a haar measure on $\mathbb{T}$, which in this case looks like $\frac{\mathrm{ln}(p)}{2\pi i} \frac{ds}{s}$ (because the contour is over a parameter between $\frac{-\pi}{\mathrm{ln}(p)} $ and $\frac{\pi}{\mathrm{ln}(p)}$.

We can write $L(\chi_s) = 1 + \chi_{s}(p) + \chi_{s}(p^2) + \cdots = \chi{s}(e) + l_p \chi_{s}(e) + l_{p^2} \chi_{s}(e) + \cdots$ where $l_p$ is translation by $p$

We may think of the function $\chi_s \mapsto \chi_{s}(e) = 1$ as a function on the set of characters restricted to $\mathbb{Z}_p^{\times}$. Inverting this, gives a function $ f_{0}(e) = \frac{\mathrm{ln}(p)}{2\pi i} \int_{\mathbb{T}} 1 \frac{ds}{s}$ just gives you the normalized measure of the circle which is $1$. Because $\chi_s(\mathbb{Z}_p^{\times})=1$, $f_0$ must be invariant under, and supported in $\mathbb{Z}_p^{\times}$

Similarly inverting $\chi \mapsto l_p(\chi)(e)$ produces function $f_{1}$ such that $l_p(f)(\mathbb{Z}_p^{\times}) =1$ and is invariant under $\mathbb{Z}_p^{\times}$. In other words, this is the characteristic function of the coset $p \mathbb{Z}_p^{\times}$. Thus $ \sum_n f_n = 1_{\mathbb{Z}_p}$