# Dirichlet series with a single zero

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) > $$\sigma_c$$.

• $1/\zeta(s){}{}$ – Wojowu May 4 '19 at 16:04
• Why are people voting to close this? – Lucia May 4 '19 at 16:54
• @Lucia I have voted to close because I thought $1/\zeta(s)$ is a (relatively) obvious counterexample, but I have not realized it relies on RH. I have retracted my vote now. – Wojowu May 4 '19 at 17:51
• Just curious: why did you need to find such an example? – KConrad May 5 '19 at 10:57
• @Jan-ChristophSchlage-Puchta $\sigma_c$ is a standard notation for the abscissa of convergence, while abscissa of absolute convergence is usually denoted with $\sigma_a$. – Wojowu May 6 '19 at 11:26

## 1 Answer

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.