We begin by reviewing Keiper and Li’s constants ${lambda_n}_{n=1,2,ldots}$, which define a real sequence whose large-order behavior sharply reflects whether the Riemann Hypothesis (RH) – a key open problem of number theory – is true or not. Tests for RH based on that sequence, such as Li’s criterion, are conceptually the most concrete and practical rewordings of RH (an appealing and interesting aspect for physicists; moreover, the analysis uses saddle-point techniques). On the dark side, the numbers $lambda_n$ have a very elusive nature; and the complexity of their numerical evaluation grows too steeply with $n$ to allow any effective access to sufficiently large $n$. Therefore, we propose new ways to make the Keiper/Li formalism simpler, more explicit and more efficient for testing RH further.

IPhT