For any arithmetical function $f(n)$, we define its derivative to be $f'(n)=f(n)\cdot \log n$ for $n\geq 1$ (see for example [1], page 45 or Wikipedia).
Fact. The functions $u(n,t)=(1/\sqrt{n})\cos (t\log n)$ and $v(n,t)=(1/\sqrt{n})\sin (t\log n)$, for integers $n\geq 1$ and real $t$, satisfy the modified Cauchy-Riemann $$\begin{cases} u_n=v_t \\ u_t=-v_n \end{cases}$$ where the derivative with respect to $n$ is taken in the sense of the derivative of an arithmetical function.
My question, and please if someone don't understand it, and another user can edit my post, or clarify my words in a comment I am agree, is
Question. Is possible (thus could be in the literature or in other case we can define it) to define a derivative $D$ acting on a complex function $f$ (this function $f$ is defined from $u$ and $v$) such that implies the system of previous modified Cauchy-Riemann equations? Or is impossible find a mixture of a discrete and continuos derivative in this way?
Thanks in advance.
References:
[1] Apostol, Introduction to Analytic Number Theory, Springer, page 45.
[2] Wikipedia, Arithmetical Function, Cauchy Riemann Equations.