Pólya’s conjecture and related sign criteria

The problem asks whether the weighted Liouville sum $f(x)$ is eventually nonnegative. This is a related sign criterion, not the original Pólya conjecture about the unweighted summatory Liouville function, which is already disproved.

Known results

• Haselgrove (1958) disproved the original Pólya conjecture; Lehman (1960) and Tanaka (1980) produced explicit counterexamples, with Tanaka’s smallest known one at $906150257$.
• For related sums $L_\alpha(x)$, logarithmic-distribution results are known conditionally on the Riemann hypothesis, linear independence, and a moment bound.
• These results do not settle eventual nonnegativity of $f(x)$.

Recent preprint: conditional sign criterion

The preprint On variants of Pólya’s conjecture proves that eventual nonnegativity of $f(x)$ would imply the Riemann hypothesis. Conversely, assuming the Riemann hypothesis, the Simple Zero Conjecture, and an additional convergence hypothesis, it derives $f(x)\sim (\log x)^3/(12\lvert\zeta(1/2)\rvert)$ and hence eventual positivity. This is a conditional advance, not an unconditional solution, and remains unverified.

Current status (as of September 2026): The original Pólya conjecture is disproved, while the stated eventual-sign question remains open; only a conditional result is reported.