$$F\\,(s) = 1/1^s + 0/2^s + 1/3^s + 1/4^s + 0/5^s + 1/6^s + 0/7^s + 1/8^s + ... $$


The coefficients of this Dirichlet series are the base-2 digits of $\sqrt2$.


Does it have an analytic continuation into the critical strip?


Assuming so, can anything be said about the locations of its poles in this region? Are there any? Are they all on the critical line?


What about $\pi$ and $e$ and $\log 2$?


Are all of these sequences "random" in the same sense as the Liouville and Moebius functions?



Dirichlet series whose coefficients are the bits of sqrt(2)
David Speyer is right. A Dirichlet series with coefficients that are random choices from 
$0,1$ does not count as a random Dirichlet series. Random choices from $-1,1$ would count as a random Dirichlet series. Taking his insight into account, I correct my prediction: With probability $1$ the Dirichlet series $\sum b\_n(x)n^{-s}$ where $b\_n(x)$ are the binary digits 
of $x \in [0,1)$ will have a meromorphic continuation to $\sigma > 1/2$ with a single simple pole at $s = 1$ with residue $1/2$. Also I now remember a reference: Armand Denjoy *L'Hypothese de Riemann sur la distribution...* C. R. Acad. Sci. Paris 192 (1931), 656-658. This paper deals with the situation where the Dirichlet series has coefficients chosen at random from $-1,1$.


