On variants of P\'olya's conjecture
In this paper, we study a Riesz-type weighted sum of the Liouville function, $f(x):=-\sum_{n\le x}\frac{\lambda(n)\log n}{\sqrt n}\log\frac{x}{n}$ for sufficiently large $x$. Motivated by a recent research on the sign criteria for the Riemann Hypothesis arising from weighted prime-counting functions, we investigate the sign behavior of $f(x)$ and its relation to the zeros of the Riemann zeta function. We first prove that if $f(x)$ is non-negative for all sufficiently large $x$, then the Riemann Hypothesis holds. The proof is based on the Mellin transform of $f$ and the analytic properties of $\frac{\zeta(2s)}{\zeta(s)}$. Conversely, assuming the Riemann Hypothesis, the Simple Zero Conjecture, and an absolute convergence condition involving the nontrivial zeta zeros, we derive an explicit formula for $f(x)$ in terms of these zeros. In particular, we show that $f(x)\sim \frac{(\log x)^3}{12|\zeta(\frac{1}{2})|}$ as $x\to \infty$. Consequently, we show that under these hypotheses, $f(x)$ is eventually positive.