I should be very grateful if someone would point out the error in the following argument, since it seems too trivial to be valid:
Let $\{pp_{n+1},pp_{n}\}$ denote the interval between prime powers, and $\rho_k$ denote the $k$th zeta zero.
The derivative of $\operatorname{li}(x)-2\Re\sum_{k=1}^{\infty}\operatorname{Ei}(\rho_k \log x)=0$ at $\{pp_{n+1},pp_{n}\}$, and since the functional equation for $\zeta(s)$ implies that $\rho_k$ necessarily comes in pairs symmetric to the line $x=\frac{1}{2}$, $\Re(\rho_k+\rho_k\ ^{'})>\frac{1}{2},$ so any pair of zeros off the critical line would surely imply that the derivative of $\operatorname{li}(x)-2\Re\sum_{k=1}^{\infty}\operatorname{Ei}(\rho_k \log x)\neq0$ at $\{pp_{n+1},pp_{n}\}?$
