AI 中文总结
本文优化了Selberg方法，通过改进算术平均值和引入半正定mollifiers，证明了Riemann zeta函数至少7%的非平凡零点位于临界线上，为该方法迄今最强结果。
AI 中文摘要
我们重新审视Zhuravlev在1974年关于Riemann zeta函数临界线上零点的Selberg符号变化方法的定量形式。Zhuravlev的工作最初以俄语发表，在西方鲜为人知，它确立了此类零点的显式正比例。利用现代技术，我们优化了该方法核心的算术平均值，该平均值对于倒数平方根系数具有精确的闭式形式，并通过一族半正定mollifiers进一步改进。由此，我们获得了Selberg方法迄今已知的最强结果，证明了Riemann zeta函数的非平凡零点中至少有$7\%$位于临界线上。
英文摘要
We revisit Zhuravlev's 1974 quantitative form of Selberg's sign-change method for the zeros of the Riemann zeta function on the critical line. Zhuravlev's work, originally in Russian and little known in the West, established an explicit positive proportion of such zeros. Using modern techniques we optimise the arithmetic mean value at the heart of the method, which has an exact closed form for reciprocal-square-root coefficients and is improved further by a positive-semidefinite family of mollifiers. We thereby obtain the strongest result yet known from Selberg's method, proving that at least $7\%$ of the non-trivial zeros of the Riemann zeta function lie on the critical line.
