A meromorphic interpolation of the harmonic numbers is given by \begin{align} H_n &= \sum_{k=1}^n \frac{1}{k} \\ &= \sum_{k=0}^{n-1} \frac{1}{k+1} \\ &= \sum_{k=0}^{n-1} \int_0^1 x^k \,\mathrm{d}x \\ &= \int_0^1 \sum_{k=0}^{n-1} x^k \,\mathrm{d}x \\ &= \int_0^1 \frac{1-x^n}{1-x} \,\mathrm{d}{x} \\ &= \gamma + \psi(n+1) \end{align} where $\psi$ is the digamma function. A meromorphic interpolation of the factorial numbers is given by the gamma function $\Gamma(n+1) = n!$. Is there a meromorphic interpolation of number-theoretic functions like the prime indicator function $\chi_\mathbb{P}$, prime-counting function $\pi$, and Möbius function $\mu$? Perhaps through the Weierstrass factorization theorem?
This question might be related (since binomial coefficients can be easily extended to continuous values).
Let $\text{sinc}(x)= \frac{\sin(\pi x)}{\pi x}$ which is entire and vanishes at every non-zero integer.
If for some $k$, $|f(n)|\le C (n!)^k $ then $$F_{k+1}(x) = \Gamma(x)^{k+1} \sum_{n=1}^\infty \frac{f(n)}{(n!)^{k+1}} \text{sinc}(x-n)$$ is analytic away from negative integers and $$ F_{k+1}(n) = f(n) , \qquad n \ge 1$$
A much more difficult question is if there exists an interpolating function not only analytic in a region but also satisfying some decay conditions, in which case it can be unique or non-existing, see Ramanujan master theorem.
Not sure if there are interpolation better suited for multiplicative functions such as $\mu(n)$. For example the inverse Mellin transform of $\Gamma(s)/\zeta(s)$ is $g(z)=\sum_{n=1}^\infty \mu(n) e^{-nz}$ and $G(u)= e^u\int_0^1 g(1+2i\pi y) e^{2i \pi u y}dy = \frac{1}{2\pi} \int_{-\infty}^\infty \frac{\Gamma(2+it)}{\zeta(2+it)} (\int_0^1 e^{(1+2i \pi y) u}(1+2i \pi y)^{-2-it}du) dt$ satisfies $G(n) = \mu(n)$