bibkey: han2026polyavariants authors: Songlin Han year: 2026 title: “On variants of Pólya’s conjecture” doi: null url: https://arxiv.org/abs/2608.27130v1 claim: “Theorem 1.1 makes eventual nonnegativity of a specified Liouville Riesz sum sufficient for RH; the reverse asymptotic in Theorem 1.2 and Corollary 1.3 also assumes simple zeros and an absolute zero-sum condition.” strata_touched: [] license: citation-only triage: anchor

A one-sided Liouville sign criterion

The primary version is arXiv:2608.27130v1, submitted 27 August 2026, 17 pages. The inspected scope is the definitions, Theorems 1.1–1.2, hypotheses H1–H3 and Corollary 1.3 on printed pages 3–4. The complete contour argument and its zero-series estimates are not independently certified here. The source is a preprint, not a Lean proof or an unconditional RH resolution. No source text is vendored.

With the Liouville function $\lambda (n)=(-1{)}^{\Omega (n)}$, the paper fixes

$$f(x)=-\sum _{n\le x}\frac{\lambda (n)\log n}{\sqrt{n}}\log \frac{x}{n}.$$

Theorem 1.1 states

$$[\exists x_0\ge 2\forall x\ge x_0:f(x)\ge 0]⟹\mathrm{RH}.$$

The converse in Theorem 1.2 uses three premises: RH, simplicity of every nontrivial zero, and

$$\sum _{\gamma >0}\frac{|\zeta (1+2i\gamma )|}{{\gamma }^2|{\zeta }^{\prime }(1/2+i\gamma )|}<\infty .$$

Under those premises Corollary 1.3 gives $f(x)\sim (\log x{)}^3/[12|\zeta (1/2)|]$ and eventual positivity. It is therefore incorrect to quote the source as establishing this converse from RH alone. The source’s Dirichlet series is $\zeta (2s)/\zeta (s)$; its pole structure and its specific logarithmic Riesz weight are part of the criterion.

For the actual FIB coefficient $e=\mu \ast \beta$, the already available Dirichlet inverse $\gamma ={\beta }^{-1}$ recovers $\mu =\gamma \ast e$. The classical square-indicator convolution $\lambda =\mu \ast 1_{\mathrm{square}}$ then identifies a possible common-source transport. Neither inverse convolution nor Riesz weighting preserves pointwise signs automatically. The formula $I_{\psi }(x)={\sum }_m{H}_m{J}_x(m)$ in FIB §397 does not identify $I_{\psi }$ with this $f$, and positivity of a single $J_x(m)$ does not verify Theorem 1.1’s quantified condition. Any application must recover the complete actual Liouville sum and prove its eventual sign.