You can read Lagarias equivalence to the Riemann's Hypothesis from this MathWorld, I am saying (5), or well from the reference [1].
Let for real numbers $0<x<1$ the factor $x^n$, where $n\geq 1$ is an integer. Then we multiply Lagarias statement by this factor and after we take the sum from $n=1$ to $\infty$. Since the Lambert series for the sum of divisor function is $\sum_{n=1}^\infty\frac{nx^n}{1-x^n}$, and a generating function for harmonic numbers is $-\frac{\log(1-x)}{1-x}$ we conclude that on assumption of the Riemann's Hypothesis $$\sum_{n=1}^\infty\frac{nx^n}{1-x^n}\leq\frac{\log(1-x)}{x-1}+\sum_{n=1}^\infty\left(e^{H_n}\log\left(H_n\right)\right)x^n.$$
You can find these generating functions for example in the corresponding Wikipedia's entries for Lambert series and Harmonic number.
Question. I believe that should be very difficult to calculate this generating function $$g(x):=\sum_{n=1}^\infty\left(e^{H_n}\log\left(H_n\right)\right)x^n$$ for $0<x<1$. But if our purpose is explore if it is possible to calculate it, what calculations can be done? That is, that I am asking, since my belief is that calculate the generating function is very difficult, is what could be the first calculations to try get it or well get a relevant property of such generating function? Summarizing, imagine that we need to study such generating function, how to start analyzing it? Thus I am asking about the first ideas, reasonings or hints. Thanks in advance.
Reference:
[1] Lagarias, An Elementary Problem Equivalent to the Riemann Hypothesis, The American Mathematical Monthly, Vol. 109, No. 6 (2002).