I've revisited an old post of mine--Dirac's Delta Functions and Riemann's Jump Function J(x) for the Primes--dealing with Riemann's "jump" or "staircase" function (aka, Π(x)) that has unit steps for each prime along the horizontal axis and smaller steps of size $1/n$ for an $n$-th power of a prime. This function is derived as an integral of the inverse Mellin transform of $\log(\zeta(1-s))$. The inverse Mellin transform can also be realized as a differential operator acting on a delta function:

$$\log[\zeta(1+xD_x)] \delta(x-1)=\sum_p \sum_{n>0}\frac{1}{n} \delta(x-p^n),$$

where the sum is over the primes $p$ and $D_x=d/dx.$

Another instance in which values of the Riemann zeta appear in a differential operator is presented in the MO-Q "Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus"

$$R_x = -\log(x)-\Psi(1+xD_x) = -\log(x) +\gamma + \sum_{n=1}^{\infty } (-1)^n\zeta (n+1)(xD_x)^n,$$

where $\gamma$ is the Euler-Mascheroni constant and $\Psi$ the digamma function. $R_x$ is an infinitesimal generator for a class of fractional calculus operators; i.e.,

$$ e^{\beta R_x} \frac{x^{\alpha}}{\alpha!}= D^\beta \frac{x^{\alpha}}{\alpha!}= \frac{x^{\alpha-\beta}}{(\alpha-\beta)!}.$$

In addition, with a the change of variable $x=e^z$, it becomes the raising operator for an Appell sequence of polynomials $p_n(z)$ related to gamma classes as shown in the MO-Q, which also has the generator

$$\frac{1}{D_z!}z^n =\exp\left [-\gamma D_z -\sum_{k=2}^{\infty } \frac{\zeta (k)D_z^k}{k} \right ]z^n = p_n(z).$$

Another example of the ocurrence of the zeta function, disguised as the Bernoulli numbers $b_n$ is a generator of the Bernoulli polynomials $B_n(x)$, which can be related to differentiation: umbrally,

$$\frac{D_x}{e^{D_x} -1} x^n = \exp(b.D_x)x^n= (b.+x)^n = B_n(x).$$

Question: In what other differential operators does the Riemann zeta function play an important role?