关于波利亚猜想的变体
On variants of Pólya's conjecture
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中文总结 AI 辅助
本文研究李奥维尔函数的里斯型加权和f(x)，证明其非负可推出黎曼假设成立，且在黎曼假设等条件下可推导f(x)的渐近式并得出其最终为正的结论。
中文摘要 AI 辅助
本文研究李奥维尔函数的里斯型加权和：对于足够大的x，定义f(x):=-∑_{n≤x} λ(n)logn / √n · log(x/n)。受近期关于加权素数计数函数引出的黎曼假设符号判据研究的启发，我们探究f(x)的符号行为及其与黎曼ζ函数零点的关系。我们首先证明：若对所有足够大的x，f(x)非负，则黎曼假设成立，该证明基于f的梅林变换及ζ(2s)/ζ(s)的解析性质。反之，假设黎曼假设、简单零点猜想及涉及非平凡ζ零点的绝对收敛条件成立，我们可推导出f(x)关于这些零点的显式公式，特别地，当x→∞时，f(x) ~ (logx)³ / (12|ζ(1/2)|)。由此，我们证明在这些假设下，f(x)最终为正。
英文摘要
In this paper, we study a Riesz-type weighted sum of the Liouville function, $f(x):=-\sum_{n\le x}\frac{λ(n)\log n}{\sqrt n}\log\frac{x}{n}$ 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 $\frac{ζ(2s)}{ζ(s)}$. 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 $f(x)\sim \frac{(\log x)^3}{12|ζ(\frac{1}{2})|}$ as $x\to \infty$. Consequently, we show that under these hypotheses, $f(x)$ is eventually positive.
