TITLE: Dirichlet series with a single zero
QUESTION [16 upvotes]: I need to find a Dirichlet series f that has the following property.
f is zero in only one point s such that Re(s) > $\sigma_c $.

REPLY [23 votes]: That such a Dirichlet series exists was a conjecture of Balazard, which was 
recently resolved by Hilberdink and Saias. If the Riemann Hypothesis is 
true, then $1/\zeta(s)$ would provide such an example (with the abscissa of conditional convergence being $1/2$), and the goal was to find an unconditional example.