If there are no mistakes in my calculations I can define for $x\geq 0$, $$f(x)=x\sum_{n=1}^\infty \frac{\mu(n)}{n^3}e^{-\frac{x}{n^3}},$$ where $\mu(n)$ is the Möbius function. And there is an expression similar to the (this was my way to define the function when I did the steps in this Wikipedia article) genuine Riesz function.
Question. How one can define the function (to extend previous real function) for complex numbers? Where can you define f(z) on $\mathbb{C}$? Thus only is required define/provide an explanation (I believe that using absolute convergence) of our function as a complex function. Thanks in advance.
I am asking it since I know that were in the literature facts about the Riesz function about its order, genus, type and distribution of zeros, and I would like to learn about it. If you can add some facts about how study these properties for our $f(z)$ you are welcome, by means comments or hints (in other case I try post a new question).
Notice that the $x$ in front of the sum plays no essential role. Looking at the series, we have that for $\Re x \ge a$
$$\left| \sum _{n \ge 1} \frac {\mu (n)} {n^3} \exp \left( - \frac x {n^3} \right) \right| \le \sum _{n \ge 1} \frac {|\mu (n)|} {n^3} \exp \left( - \frac {\Re x} {n^3} \right) \le \sum _{n \ge 1} \frac 1 {n^3} \exp \left( - \frac a {n^3} \right) \dots$$
Since $\exp \left( - \frac a {n^3} \right) \to 1$ for $n \to \infty$, there exist $n_0$ such that for $n \ge n_0$ we have $\exp \left| \left( - \frac a {n^3} \right) \right| \le 2$, so the above is
$$\dots \le \sum _{n \ge 1} \frac 1 {n^3} 2 = 2 \zeta (3) .$$
This shows that the series which gives $f(x) / x$ converges uniformly on the half-planes of the form $\Re z \ge a$ for every $a \in \Bbb R$. In particular, this shows that this series converges uniformly on every compact, and since the partial sums are entire, and analyticity is preserved by compact convergence, the sum of the series will also be entire, and so will be $f$.