Background: Bombieri and Lagarias showed that a function $f$ with roots $\rho=x+iy$ satisfies has all its roots lying on $x=\frac12$ if and only if
$$\lambda_n :=\sum_\rho 1-\left(1-\frac{1}{\rho}\right)^n$$
is nonnegative for all $n$.
When $f$ is the Riemann zeta restricted to the critical strip, this gives an equivalent statement for the Riemann Hypothesis: find a (sharp, I think) lower bound for $\lambda_n$.
Question: Is there a simple function that serves as an upper bound for $\lambda_n$ for all $n$?
Motivation: I was trying to answer this question, and I suspected there would be a nice way to say that the question was hard because it implies the Riemann Hypothesis. I managed to find another way to show the implication, but I would still like to complete this approach.
This formulation of $\lambda_n$ suggests an easy continuous extension to $\lambda:\Bbb R_+\to\Bbb R$. However, for the purposes of the equivalence, I wanted a function which never went above 1, so if we had some upper bound $\lambda\leq B$, then I could use $f(x)=\lambda(x)/B(x)$ [and so $\lfloor f\rfloor$ in the question would be nonzero iff RH.]
However, I'm a completely new to analytic number theory and so I have absolutely no intuition about these numbers. I used WolframAlpha to plot the first two terms in the partial sum, and I played around with it a little bit, so I am pretty sure that $\lambda(x)$ is real. But that's about it.