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[图片，科普，数学] 素数（39）：黎曼ζ函数 Riemann zeta function 的零点之二

已有 221 次阅读 2026-4-19 20:34 |个人分类:资料与科普|系统分类:科研笔记

Yet, in fact, as I shall show here with very good reasons, the properties of the numbers known today have been mostly discovered by observation, and discovered long before their truth has been confirmed by rigid demonstrations.

——Leonhard Paul Euler

事实上，正如我以非常充分的理由在此将要指出的那样，今天人们所知道的数的性质，几乎都是由观察所发现的，并且早在用严格论证确认其真实性之前就被发现了。

[图片，科普，数学] 素数（39）：黎曼ζ函数 Riemann zeta function 的零点之二

黎曼ζ函数： Riemann zeta function, Riemann ζ function

平凡零点： trivial zero

非平凡零点： nontrivial zero

临界带： critical strip

素数： prime number

算术基本定理： fundamental theorem of arithmetic

素数计数函数： prime counting function

素数定理： prime number theorem

对数积分： logarithmic integral

唯一分解定理： unique factorization theorem

黎曼假设： Riemann Hypothesis

希尔伯特的第 8问题： Hilbert's 8th Problem

网上有不少对黎曼ζ函数（Riemann zeta function, Riemann ζ function）零点（zeros）数值计算的结果。陆续搜集汇总如下。

下面是 2015年文献的一个数值计算结果：

Figure 1 ps519618f1_hr.jpg

图1 Figure 1 ps519618f1_hr.jpg

https://content.cld.iop.org/journals/1402-4896/90/10/108015/revision1/ps519618f1_hr.jpg

Figure 1. Newton flow of the Riemann zeta function in the complex plane given by the argument s ≡ σ+iτ in the neighborhood of the critical line σ =1/2, indicated by the orange line.

图1. 黎曼ζ函数在复平面内的牛顿流，由临界线 σ =1/2 附近的自变量 s ≡ σ+iτ 给出，由橙线表示。

Figure 2 ps519618f2_hr.jpg

图2 Figure 2 ps519618f2_hr.jpg

https://content.cld.iop.org/journals/1402-4896/90/10/108015/revision1/ps519618f2_hr.jpg

Figure 2. Lines in the complex plane where the Riemann zeta function ζ is real (green) depicted on a relief representing the positive absolute value of ζ for arguments s ≡ σ+iτ where the real part of ζ is positive, and the negative absolute value of ζ where the real part of ζ is negative.

图2. 复平面中的线，其中黎曼ζ函数ζ为实数（绿色），描绘在浮雕上，代表参数 s ≡ σ+iτ 的ζ的正绝对值，其中ζ的实数部分为正，ζ的实部为负。

Figure 3 ps519618f3_hr.jpg

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Figure 3. Newton flow in the complex plane together with the lines of constant height Ck ≡ |ζ| =1 (violet curves) and lines Rk (green solid curves) where ζ is real analyzed in [7].

图3. 复平面中的牛顿流，以及恒定高度Ck≡|ζ|=1的线（紫色曲线）和线Rk（绿色实心曲线），其中ζ是实数，在[7]中进行了分析。

Figure 4 ps519618f4_hr.jpg

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https://content.cld.iop.org/journals/1402-4896/90/10/108015/revision1/ps519618f4_hr.jpg

Figure 4. Newton flow of ζ on the Riemann sphere in the neighborhood of the pole of ζ indicated by the black dot (right picture), and of the first trivial zero, marked by the red dot closest to the South pole (left picture).

图4. ζ在黎曼球上的牛顿流，位于由黑点表示的ζ极点附近（右图），以及由最靠近南极的红点标记的第一个平凡零点附近（左图）。

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Figure 5. Newton flow of ζ on the Northern hemisphere of the Riemann sphere where the North pole serves as a source. The green line in the form of a cardioid emerging from the pole to the right and running mostly on the edge of the globe is the separatrix already shown in figure 4.

图5. ζ在黎曼球北半球的牛顿流，其中北极作为源。图4中已经显示了一条从极点向右延伸并主要在地球边缘延伸的心形绿线。

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Figure 6. Newton flow of ζ close to the North pole of the Riemann sphere along the positive critical line. On the green and blue lines ζ assumes real values.

图6. ζ的牛顿流沿着正临界线靠近黎曼球的北极。在绿线和蓝线上，ζ取实际值。

Figure 7 ps519618f7_hr.jpg

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Figure 7. Newton flow of the function f defined by equation (22), which is known [3] not to satisfy the Riemann hypothesis but is otherwise very close to ζ.

图7. 由方程（22）定义的函数f的牛顿流，已知[3]不满足黎曼假设，但在其他方面非常接近ζ。

Figure A1 ps519618f8_hr.jpg

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Figure A1. Proof that all zeros of the sine function S lie on the real axis solely based on the asymptotic properties of (i) the corresponding Newton flow represented in the complex plane (center), and (ii) the phases of the individual flow lines (top and bottom).

图A1. 仅基于（i）复平面（中心）中表示的相应牛顿流和（ii）单个流线（顶部和底部）的相位的渐近性质，证明正弦函数S的所有零点都位于实轴上。

Figure A2 ps519618f9_hr.jpg