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[图片，科普，数学] 素数（40）：黎曼假设与 end effects 端点效应之二

已有 222 次阅读 2026-4-20 21:26 |个人分类:资料与科普|系统分类:科普集锦

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（40）：黎曼假设 Riemann Hypothesis 与 end effects 端点效应之二

—— 黎曼假设 Riemann Hypothesis、黎曼素数计数函数 Riemann Prime Counting Function，在进行数值计算时，也会遇到“端点效应 end effects”吗？这是“周期性分解”的共同问题吗？

端点效应： end effects, endpoint effect

边缘效应： edge effects, border effects, boundary effects

虚假的波动： fictional waves

黎曼素数计数函数： Riemann Prime Counting Function

傅里叶展开： Fourier expansion

小波分解： wavelet decomposition

经验模式分解： empirical mode decomposition

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

图1 Me4eT.png,

黎曼假设，黎曼ζ函数 Riemann zeta function 的零点

https://i.sstatic.net/Me4eT.png

There is only the Riemann hypothesis open, whether there are infinitely many zero on the critical line Re(s) = 1/2.

It is possible to transform the average line approximating, the Mangoldt lambda function, into the stair plot of the number of positive primes, in Mathematica this is the PrimePi function, with the Riemann ζ zeros so that the stairs are reproduced exactly. But this is an infinity process. For finite numbers of Riemann ζ zeros used, the approximation remains undulated with high deviation at the steps.

【机器翻译】在临界线 Re(s) = 1/2 上是否有无穷多个零，只有黎曼假设是开放的。

有可能将平均线近似，即 Mangoldt lambda 函数，转换为正素数数量的阶梯图，在 Mathematica 中，这是 PrimePi 函数，黎曼ζ为零，从而精确地再现阶梯。但这是一个无限的过程。对于使用的有限个黎曼ζ零点，近似值在步长处保持波动，偏差很大。

图2 page5455.jpg,

黎曼的素数计数函数 prime counting function 估计 R(x)

page 55 of Hans Riesel's book

https://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/page5455.jpg

An excellent illustration of this is provided on page 55 of Hans Riesel's book, which compares π(x) with

R10 page 55 of Hans Riesel's book,_放大_后期.jpg

i.e. the Riemann function corrected by contributions from the first 10 pairs of nontrivial zeta zeros. Zagier's article also includes graphs of the first few Tk(x) as well as R₁₀(x) and R₂₉(x).

【机器翻译】Hans Riesel的书第55页对此进行了很好的说明，该书将 π(x) 与

即通过前10对非平凡ζ零点的贡献校正的黎曼函数。Zagier的文章还包括前几个 Tk(x) 以及 R₁₀(x) 和 R₂₉(x) 的图。