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 $\varphi (n)$?
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.
Peter provides further clarification in another comment.
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 $a(n)=\varphi(n)$