发表机构
University of Vienna（维也纳大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究黎曼-冯·曼戈尔特素计数函数显式公式的收敛性，证明非平凡零点部分和非O(T^θ)、∑_ρ R(x^ρ)发散，且平凡零点贡献也发散。
AI 中文摘要
素计数函数π(x)的显式公式通常归功于黎曼和冯·曼戈尔特，其核心表述为方程π(x)=R(x)-∑_ρ R(x^ρ)，其中求和遍历黎曼ζ函数的所有零点ρ，非平凡零点按其虚部绝对值递增排序且计重数。这尤其包含如下论断：非平凡零点的部分和ΣR_T(x):=∑_{0<|Im(ρ)|≤T} R(x^ρ)在T→∞时收敛。记Θ:=sup{Re(ρ):ζ(ρ)=0,0<Re(ρ)<1}（近期被称为“黎曼常数”），我们证明：对每个固定的x>1及每个θ<Θ，和式ΣR_T(x)都不是O(T^θ)。由此可得limsup_{T→∞}|ΣR_T(x)|=∞，且∑_ρ R(x^ρ)发散。我们在文末证明，一种适配但更简单的策略也能得出平凡零点对∑_ρ R(x^ρ)的贡献发散的结论。
英文摘要
An explicit formula for the prime-counting function $π(x)$, usually attributed to Riemann and von Mangoldt, is prominently stated as the equation $π(x)=R(x)-\sum_ρR(x^ρ)$, where the sum runs over all zeros $ρ$ of the Riemann $ζ$-function, the non-trivial ones being ordered by increasing absolute value of their imaginary parts and counted with multiplicity. This particularly entails the claim that the partial sums over the non-trivial zeros, $ΣR_T(x):=\sum_{0<|\Im m(ρ)|\le T} R(x^ρ)$ converge as ${T\to\infty}$. Writing $Θ:=\sup\{\Re e(ρ):\ ζ(ρ)=0,\ 0<\Re e(ρ)<1\},$ for what has recently been called ``Riemann's constant'', we prove that, for every fixed $x>1$ and every $θ<Θ$, the sums $ΣR_T(x)$ are not $O(T^θ)$. As a consequence, $\limsup_{T\to\infty}|ΣR_T(x)|=\infty$ and $\sum_ρR(x^ρ)$ diverges. We conclude the paper by showing that an adapted, but simpler strategy also gives the analogous result -- and hence divergence -- for the contribution of the trivial zeros to $\sum_ρR(x^ρ)$. In both cases the exponent $Θ$ is proved to be sharp.
CommentsRevised and substantially strengthened version. We establish sharp upper bounds, and hence the exact power exponent, for the partial sums over both the non-trivial and the trivial zeros. We also deal with the divergence when all zeros $ρ$ are summed jointly in increasing order of $\lvertρ-\tfrac12\rvert$
