Riemann-Siegel 余项作为分数求和项
The Riemann-Siegel remainder as a fractional summand
浏览论文内容
中文总结 AI 辅助
本文通过变量替换将 Riemann-Siegel 余项分解为两个分数部分求和项，证明 ζ 可表示为 Dirichlet 和加分数余项，并在临界线上验证了对称性，且余项近似周期为 1。
中文摘要 AI 辅助
从 Siegel 1932 年论文中给出的 ζ(s) 的 Riemann-Siegel 分解出发，我们引入变量替换 t=I(T)，将 Siegel 的耦合对“虚部 t / 求和指标 m”替换为一个实指标 T。然后我们将 Siegel 的单个余项积分 R 分解为两个精确部分 R_{1ps} 和 R_{2ps}，并证明 R=R_{1ps}+R_{2ps}，从而 ζ=Σ_1+R_{1ps}+Σ_2+R_{2ps}。我们的核心观察是，这些余项中的每一个都不过是附加到其 Dirichlet 和上的一个额外的分数部分求和项：ζ(s)=∑_{n=1}^{m}n^{-s}+\hat{d}_1(m+1)^{-s}+χ(s)∑_{n=1}^{m}n^{s-1}+\hat{d}_2χ(s)(m+1)^{s-1}，其中 \hat{d}_1,\hat{d}_2 是实数（在临界线上始终为正），即这两个求和项所使用的分数。作为推论，当 σ=1/2 时，有 d_1=d_2（等价地 \hat{d}_1=\hat{d}_2）；这一事实已在 Lean 中正式验证。此外，通过这种重新缩放，余项在 T 中近似以 1 为周期，并在 T 的分数部分收敛到一个固定波形。Siegel 的 R 的这一分解是通过实验数学发现的，使用了部分和的螺旋可视化，文中稍后描述。我们还讨论了其他一些观察结果，包括我们称之为阴阳曲线、零点计数函数以及等长腿轨迹的椭圆。
英文摘要
Starting from the Riemann-Siegel decomposition of $ζ(s)$ given in Siegel's 1932 paper, we introduce a change of variable, $t=I(T)$, that replaces Siegel's coupled pair "imaginary part $t$ / summation index $m$" with one real index $T$. We then split Siegel's single remainder integral $R$ into two exact pieces, $R_{1ps}$ and $R_{2ps}$, and prove that $R=R_{1ps}+R_{2ps}$, so that $ζ=Σ_1+R_{1ps}+Σ_2+R_{2ps}$. Our central observation is that each of these remainders is nothing more than one additional, fractional partial summand appended to its Dirichlet sum: $ζ(s)=\sum_{n=1}^{m}n^{-s}+\hat{d}_1(m+1)^{-s}+χ(s)\sum_{n=1}^{m}n^{s-1}+\hat{d}_2χ(s)(m+1)^{s-1}$, with $\hat{d}_1,\hat{d}_2$ real numbers (always positive on the critical line), the fractions of those two summands that are used. As a corollary, when $σ=\frac{1}{2}$ one has $d_1=d_2$ (equivalently $\hat{d}_1=\hat{d}_2$); this fact is formally verified in Lean. Also, with this rescaling the remainder terms are nearly periodic in $T$ with period one, converging to a fixed waveform in the fractional part of $T$. This decomposition of Siegel's $R$ was discovered through experimental mathematics using a spiral visualization of the partial sums, described later in the paper. We also discuss a number of other observations, including what we call the yin yang curves, the zero counting function, and ovals of equal length leg loci.
