Analytic representation of Euler's totient $\varphi (n)$ Let
$$\pi_0 (x)=\frac12 \lim_{h \to 0}[\pi (x+h)+\pi (x-h)]$$
where $\pi (x)$ is the prime counting function. Then
$$\pi_0 (x)=\sum_{n=1}^\infty \frac{\mu (n)}{n}f(x^{1/n})$$
where
$$f(x)=\operatorname{li}(x)-\sum_{\rho} \operatorname{li}(x^\rho)-\log (2)+\int_x^\infty \frac{dt}{t(t^2-1)\log t},$$
and the sum is over the non-trivial zeros $\rho$ of the Riemann zeta function. This is taken from https://en.wikipedia.org/wiki/Explicit_formulae_for_L-functions.
Euler's totient function $\varphi (n)$ is a number-theoretic function just as $\pi (n)$. Is there an analogous, analytic representation of $\varphi (n)$?
 A: The totient summatory function
$$\Phi(x)=\sum\limits_{n=1}^x\varphi(n)$$
associated with
$$\frac{\zeta(s-1)}{\zeta(s)}=\sum\limits_{n=1}^\infty\varphi(n)\,n^{-s},\quad\Re(s)>2$$
is analogous to the second Chebyshev function
$$\psi(x)=\sum\limits_{n=1}^x\Lambda(n)$$
associated with
$$\frac{\partial\log\zeta(s)}{\partial s}=-\frac{\zeta'(s)}{\zeta(s)}=\sum\limits_{n=1}^\infty\Lambda(n)\,n^{-s},\quad\Re(s)>1$$
and the Riemann prime-power counting function
$$\Pi(x)=\sum\limits_{n=2}^x\frac{\Lambda(n)}{\log(n)}$$
associated with
$$\log\zeta(s)=\sum\limits_{n=2}^\infty\frac{\Lambda(n)}{\log(n)}\,n^{-s},\quad\Re(s)>1$$
The analytic representation of $\pi_0(x)$ is based on the analytic representation for $\Pi_0(x)$ and the relationship $\pi(x)=\sum\limits_{n=1}^\infty\frac{\mu(n)}{n}\,\Pi\left(x^{\frac{1}{n}}\right)$ which is the Möbius inversion of the relationship $\Pi(x)=\sum\limits_{n=1}^\infty\frac{1}{n}\,\pi\left(x^{\frac{1}{n}}\right)$.
But an analytic formula for $\Phi(x)$ would perhaps yield an analytic formula for $\varphi(n)$ since $\varphi(n)=\Phi(n)-\Phi(n-1)$. For example, I believe $\Lambda(n)=\psi_o\left(x+\frac{1}{2}\right)-\psi_o\left(x-\frac{1}{2}\right)$ evaluates correctly at integer values of $x\ge 2$ since the explicit formula for $\psi_o(x)$ only converges for $x>1$.
The problem is I don't believe the analytic formula for $\Phi(x)$ at the referenced  Wikipedia article converges.
PeterHumphries explains the reason for this in a comment on my my related question.

The explicit formula for $\sum_{n \leq x} \varphi(x)$ is wrong; when you shift the contour, the shifted contour integral is not small. One can use this to show that the error term for this sum is at least as large as a constant multiple of $x\sqrt{\log \log x}$ infinitely often.

Peter provides further clarification in another comment.

What I mean is that $\sum_{n \leq x} \varphi(x)$ has a main term, coming from the pole of $\zeta(s - 1)/\zeta(s)$ at $s = 2$, and an error term of size at least $x\sqrt{\log \log x}$, which does not come from the poles of $\zeta(s - 1)/\zeta(s)$ at the zeroes of $\zeta(s)$.

Note that in the two comments quoted above $\varphi(x)$ should have been $\varphi(n)$.
I've investigated a potential analytic representation of $\varphi(n)$ based on this answer I posted to a question about an entire function interpolating the Möbius function $\mu(n)$, but it wouldn't really be analogous to explicit formulas related to the Riemann prime-power counting function $\Pi(x)$ (referred to as $f(x)$ in the question above) or the second Chebyshev function $\psi(x)$.
In my answer linked in the paragraph above formulas (5) and (6) are only equivalent to formulas (7) and (8) when $F_a(s)=\sum\limits_{n=1}^\infty \frac{a(n)}{n^s}$ converges for $\Re(s)\ge 2$ which it doesn't in the case of $a(n)=\varphi(n)$ where $F_a(s)=\frac{\zeta(s-1)}{\zeta(s)}$. I believe as $K\to\infty$ formula (10) for $\tilde{a}(s)$ may still be valid at integer values of s when $|s|\le K$, but perhaps generally diverges at non-integer values of $s$ because this condition isn't met.
The following figure illustrates formula (10) for $\tilde{a}(s)$ corresponding to $a(n)=\varphi(n)$ where formula (10) is evaluated at $K=20$ and $f=1$. The red discrete portion of the plot represents the evaluation of $\varphi(|s|)$ at integer values of $s$. Note formula (10) for $\tilde{a}(s)$ converges to $0$ at $s=0$ and to $a(|s|)$ at positive and negative integer values of $s$.


Figure (1): Illustration of formula (10) for $\tilde{a}(s)$ corresponding to $a(n)=\varphi(n)$
