Consider the Dirichlet series (for fixed $0 < a \leq 1$):
$$\zeta_a(s) = \sum_{n\geq 1}\frac{a^n}{n^s}$$
which reduces to the Riemann zeta function for $a=1$. What is known about this function, in terms of functional equations or relations to other standard Dirichlet series? If we try to do the standard trick with Mellin transforms to relate this to a Jacobi theta function, it works and we get that: $$\pi^{-s/2}\Gamma(s/2)\zeta_a(s) = \int_0^\infty\frac{\nu(z,iy)-1}{2}y^{s/2}\frac{dy}{y}$$ for $\nu(z,\tau)$ the Jacobi theta function and $\exp(2\pi iz) = a$: $$\nu(z,\tau) = \sum_{n\in \mathbb Z}\exp(\pi in^2\tau + 2\pi inz).$$ Is there some nice functional equation with non zero $z$ for $\nu(z,\tau)$?
The motivation for this post comes from this blog post of Matt baker where he shows that for $a$ integral, $\sum_{d|n}\mu(n/d)a^d \equiv 0 \pmod n$ as a generalization of Fermat and Euler's theorem.
This suggests that the arithmetic function $n \to a^n$ has interesting arithmetical properties and perhaps the corresponding Dirichlet function would be interesting. Unfortunately, $\zeta_a(s)$ is divergent for $a > 1$, the regime we actually care about!
So perhaps, we should instead look for p-adic analogues. More precisely, for $0 < a < 1$ a rational number such that $a-1$ is a p-adic unit. I believe I can show in this case that for $n$ a negative integer, $\zeta_a(n)$ takes rational values that are moreover p-adically integral and that there is a p-adic analytic function that interpolates these values.
Have these functions been studied before?
