[原创]k生素数群的数量公式

白新岭

重生888@ 发表于 2021-5-25 07:36
白先生好！您在二楼叹息除哈-李渐进公式，没有其他渐进公式。我现在的新新公式是计算哥猜的真正渐进公式！ ...

你的到来可能使本帖升温。截止2021年5月25日周二10:22分，温度95°，上升了1°。

重生888@

白新岭 发表于 2021-5-25 10:23
你的到来可能使本帖升温。截止2021年5月25日周二10:22分，温度95°，上升了1°。

祝贺您的帖子热度升温！我的新新公式克服了呆板不变的系数，用斐波那契数列倒数和做辅助系数，有了渐进功能。使用它可以越来越收敛，不会无限扩大；素数定理越来越吻合，因此，不会出现异常！谢谢！

白新岭

连续素数之间的间隔：
介绍：
The computation of the first occurrence of prime gaps of a given (even) size between consecutive primes has some theoretical interest [1, problem A8], [2]. Let p_k be the k-th prime number, i.e., p_1=2, p_2=3, p_3=5, ..., and let g_k=p_(k+1)-p_k be the gap between the consecutive primes p_k and p_(k+1). Harald Cramér [3] conjectured, based on probabilistic ideas, that the large values of g_k grow like (log p_k)^2. Our empirical data does not allow us to discriminate between this growth rate and, for example, (log pi(p_k))^2, where pi(x) is the usual prime counting function (note that pi(p_k)=k). Furthermore, the bounds presented below suggest yet another growth rate, namely, that of the square of the so-called Lambert W function. These growth rates differ by very slowly growing factors (like log log p_k). Much more data is needed to verify empirically which one is closer to the true growth rate.

Let P(g) be the least prime such that P(g)+g is the smallest prime larger than P(g). The values of P(g) are bounded, for our empirical data, by the functions
计算连续素数之间第一次出现给定(偶数)大小的素数间隔，有一些理论上的意义[1，问题 a8] ，[2]。设 p _ k 为 k 次素数，即 p _ 1 = 2，p _ 2 = 3，p _ 3 = 5，... ，且 g _ k = p _ (k + 1)-p _ k 为连续素数 p _ (k + 1)之间的间隔。Harald cramér [3]根据概率论的观点推测，g _ k 的大值会像(log p _ k) ^ 2那样增长。我们的经验数据不允许我们区分这个增长率和，例如，(log pi (p_k)) ^ 2，其中 pi (x)是通常的素数计数函数(注意 pi (p_k) = k)。此外，下面列出的界限表明还有另一个增长率，即所谓的朗伯W函数平方增长率。这些生长速率的差异由非常缓慢的生长因素(如对数 loglogp_k)。需要更多的数据来验证哪一个更接近真实的增长率。让 p (g)是最小素数，使得 p (g) + g 是大于 p (g)的最小素数。对于我们的经验数据，p (g)的值是有界的
这应该与熊一兵先生研究的边界有关。

白新岭

                         0.5                                     0.5
                  0.5   g                                 0.5   g
P_min(g) = 0.12  g     e        and    P_max(g) = 30.83  g     e  
这是给出的计算公式（网上数据，没有亲自验证）

白新岭

For large g, there bounds are in accord with a conjecture of Marek Wolf [4]
对于大 g，其界限与马雷克・沃尔夫的一个猜想一致[4]

翻译来自彩云小译

白新岭

Computational results
Our prime gap computations are a by-product of our Goldbach conjecture verification effort [5]. We have computed the gaps between all prime numbers smaller than 4·10^18, and, so far, double-checked our results up to 4·10^17 (see the detailed status of this extensive computation). Our results agree with whose presented in [6], [7], [8], and [9]. In this table [14KiB, compressed with gzip] we present the values of P(g) we were able to compute, as well as counts of the number of times each gap occurred. The record-holders, i.e., numbers larger than all previous ones of the same kind, are clearly marked in the table. The following figure presents a graph with the available values of P(g). The black line represents the lower bound for P(g) suggested by Cramér's conjecture.
计算结果素数间隙计算是我们的哥德巴赫猜想验证工作的副产品[5]。我们计算了所有小于410 ^ 18的质数之间的间隔，到目前为止，我们反复检查了结果，直到410 ^ 17。我们的结果与文献[6] ，[7] ，[8]和[9]中的结果一致。在这个表[14kib，用 gzip 压缩]中，我们给出了我们能够计算的 p (g)的值，以及每个间隙出现的次数。记录保持者，也就是比以前所有同类数字都大的数字，在表格中清楚地标明。下图展示了一个 p (g)可用值的图表。黑线代表克拉门猜想提出的 p (g)的下界。

白新岭

The two kinds of record-holders marked in the table mentioned above correspond to the values of P(g) closer to one of the two bounds presented in the introduction. In this figure it can be seen that the first occurrence of a prime gap of 1132 is abnormally small.

Let D(x;g) be the relative frequency of occurrence of the gap g for all prime numbers not larger than x. The following figure presents a graph of this function, computed for our current verification limit of the Goldbach conjecture.
上表中标出的两种记录持有人的数值相当于 p (g) ，接近导言中提出的两个界限中的一个。在这张图中可以看出，1132的素数间隙的第一次出现是异常小的。 d (x; g)是所有素数出现的相对频率。下图展示了这个函数的一个图形，是为我们目前对哥德巴赫猜想的验证极限而计算的。

白新岭

The approximate exponential decay of D(x;g) is explained in [10]. It is visible in this figure that the values of D(x;g) when g is a multiple of 6=2·3 (white dots) are larger than those of their neighbors (yellow dots); those of the multiples of 30=2·3·5, and of 210=2·3·5·7, are even larger
D (x; g)的大约指数衰减在[10]中有解释。从图中可以看出，当 g 是6 = 23(白点)的倍数时，d (x; g)的值大于它们的邻居(黄点) ，而30 = 235和210 = 2357的倍数则更大

白新岭

Hardy-Littlewood constants and jumping champions
If one assumes the truth of the prime k-tuple conjecture [11], it is possible to estimate the number of occurrences up to x of a prime gap of g, denoted here by N(x;g), for relatively small values of g [5], [12]. With the help of Siegfried Herzog we managed to compute all relevant constants necessary to do this for all (even) g smaller than 214. (We double-checked each other's results in full up to g=190 and I checked some of Zig's results for larger values of g.) The constants, and minimal details about how they can be used to estimate N(x;g), can be found here [97KiB, compressed with gzip].
哈代-利特伍德常数和跳跃冠军赛如果一个人假设素数 k-tuple 猜想[11]是正确的，那么对于 g [5] ，[12]这个相对较小的值，可以估计出 g 的素数间隙 x 的出现次数。在 siegfried herzog 的帮助下，我们设法计算了所有小于214的 g 所需的相关常数。(我们反复检查了对方的结果，直到 g = 190，我检查了 zig 的一些结果，看是否有更大的 g 值)这些常量，以及如何使用它们来估计 n (x; g)的最小细节，可以在这里找到[97kib，压缩与 gzip ]。

白新岭

It was observed empirically that these estimates of N(x;g) appear to not deviate from their true values by more that

sqrt(2 N(x;g) log log N(x;g))
(law of the iterated logarithm). Using a tolerance 100 times larger than this, our data suggests that:
根据经验观察，这些估计 n (x; g)似乎没有偏离他们的真实值更多的 sqrt (2 n (x; g) log log n (x; g))(重对数律)。使用100倍于此的容差，我们的数据表明: