[原创]k生素数群的数量公式

白新岭 · 发表于 2021-7-8 17:39

本帖最后由白新岭于 2021-7-8 17:41 编辑

似乎Jiangchunxuan 在歪曲事实，哈代-李的公式前系数明明是：∏\({P^{k-1}*(P-v(P)）}\over(P-1)^k\),他愣是把间距>2P时，v（P）=k，改成了v（P）=k-1，他的是这个值，不过他做了恒等变形：P-1-χ（P），即提前去了1.
实际上，哈代-李的v（P）=k（距离>2P时），实际上是k生素数的素数式（0,2n1,2n2，....2n（k-1）），k个偶数的同余个数。

白新岭 · 发表于 2021-7-8 17:50

C_2 = Product_{ p prime > 2} (p * (p-2) / (p-1)^2) is the 2-tuple case of the Hardy-Littlewood prime k-tuple constant (part of First H-L Conjecture): C_k = Product_{ p prime > k} (p^(k-1) * (p-k) / (p-1)^k).
C _ 2 = 乘积 _ { p 素数 > 2}(p * (p-2)/(p-1) ^ 2)是 hardy-littlewood 素数 k-tuple 常数(第一 h-l 猜想的一部分)的二元情形: c _ k = 乘积{ p 素数 > k }(p ^ (k-1) * (p-k)/(p-1) ^ k)。
https://www.so.com/link?m=bf20Mb ... iL0jc1RB7POjfKLk%3D
在这个连接中打开即可（A005597）

白新岭 · 发表于 2021-7-8 17:56

从上楼的英文中和翻译成的汉语可知，哈代-李特伍尔德的k元组素数的公式中的系数并不是Jiang chunxuan所批判的那样，他在歪曲事实，给自己脸上贴金，明明哈代-李的正确，他把原来的给改了，变成（P-k+1），说是false的，而把自己变成哈代-李的原来面貌，说成自己的，他在造假，歪曲事实。

白新岭 · 发表于 2021-7-8 17:58

Although C_2 is commonly called the twin prime constant, it is actually the prime 2-tuple constant (prime pair constant) which is relevant to prime pairs (p, p+2m), m >= 1.

The Hardy-Littlewood asymptotic conjecture for Pi_2m(n), the number of prime pairs (p, p+2m), m >= 1, with p <= n, claims that Pi_2m(n) ~ C_2(2m) * Li_2(n), where Li_2(n) = Integral_{2, n} (dx/log^2(x)) and C_2(2m) = 2 * C_2 * Product_{p prime > 2, p | m} (p-1)/(p-2), which gives: C_2(2) = 2 * C_2 as the prime pair (p, p+2) constant, C_2(4) = 2 * C_2 as the prime pair (p, p+4) constant, C_2(6) = 2* (2/1) * C_2 as the prime pair (p, p+6) constant, C_2(8) = 2 * C_2 as the prime pair (p, p+8) constant, C_2(10) = 2 * (4/3) * C_2 as the prime pair (p, p+10) constant, C_2(12) = 2 * (2/1) * C_2 as the prime pair (p, p+12) constant, C_2(14) = 2 * (6/5) * C_2 as the prime pair (p, p+14) constant, C_2(16) = 2 * C_2 as the prime pair (p, p+16) constant, ... and, for i >= 1, C_2(2^i) = 2 * C_2 as the prime pair (p, p+2^i) constant.
虽然 c2通常被称为孪生素数常数，但它实际上是与素数对(p，p + 2 m)相关的素数2元组常数(素数对常数) ，m > = 1。Pi _ 2m (n)的 hardy-littlewood 渐近猜想，素数对(p，p + 2 m)的个数，m > = 1，p < = n，声称 pi _ 2m (n) ~ c _ 2(2m) * li _ 2(n) ，其中 li _ 2(n) = {2，n }(dx/log ^ 2(x))和 c _ 2(2m) = 2 * c _ 2 * 乘积{ p prime > 2，p | m }(p-1)/(p-2) ,得到: c2(2) = 2 * c2为素对(p，p + 2)常数，c2(4) = 2 * c2为素对(p，p + 4)常数,c2(6) = 2 * (2/1) * c2为素对(p，p + 6)常数，c2(8) = 2 * c2为素对(p，p + 8)常数,c2(10) = 2 * (4/3) * c2为素对(p，p + 10)常数，c2(12) = 2 * (2/1) * c2为素对(p，p + 12)常数,c2(14) = 2 * (6/5) * c2为素对(p，p + 14)常数，c2(16) = 2 * c2为素对(p，p + 16)常数，. 。对于 i > = 1，c _ 2(2 ^ i) = 2 * c _ 2作为素数对(p，p + 2 ^ i)常数。

白新岭 · 发表于 2021-7-8 18:00

C_2 also occurs as part of other Hardy-Littlewood conjectures related to prime pairs, e.g., the Hardy-Littlewood conjecture concerning the distribution of the Sophie Germain primes (A156874) on primes p such that 2p+1 is also prime.

Another constant related to the twin primes is Viggo Brun's constant B (sometimes also called the twin primes Viggo Brun's constant B_2) A065421, where B_2 = Sum (1/p + 1/q) as (p,q) runs through the twin primes.

Reciprocal of the Selberg-Delange constant A167864. See A167864 for additional comments and references. - Jonathan Sondow, Nov 18 2009

C_2 = Product_{prime p>2} (p-2)p/(p-1)^2 is an analog for primes of Wallis' product 2/Pi = Product_{n=1 to oo} (2n-1)(2n+1)/(2n)^2. - Jonathan Sondow, Nov 18 2009

One can compute a cubic variant, product_{primes >2} (1-1/(p-1)^3) = 0.855392... = (2/3) * 0.6601618...* 1.943596... by multiplying this constant with 2/3 and A082695. - R. J. Mathar, Apr 03 2011

Cohen (1998, p. 7) referred to this number as the "twin prime and Goldbach constant" and noted that, conjecturally, the number of twin prime pairs (p,p+2) with p <= X tends to 2*C_2*X/log(X)^2 as X tends to infinity. - Artur Jasinski, Feb 01 2021
C _ 2也出现在其他与素数对有关的 hardy-littlewood 猜想的一部分，例如，hardy-littlewood 猜想关于素数 p 上 sophie germain 素数(a156874)的分布，这样2p + 1也是素数。另一个与孪生素数有关的常数是 viggo brun 的常数 b (有时也称为孪生素数 brun 的常数 b _ 2) a065421，其中 b _ 2 = sum (1/p + 1/q) as (p，q)贯穿于孪生素数 a _ 167864的倒数。167864 for additional comments and references.- jonathan sondow，nov 182009c2 = product _ { prime p > 2}(p-2) p/(p-1) ^ 2是 wallis’ product 2/pi = product _ { n = 1 to oo }(2n-1)(2n + 1)/(2n) ^ 2的素数类似物。- jonathan sondow，2009年11月18日1可以计算一个立方变量，乘以2/3和 a082695，乘以这个常数。- r.j. mathar，apr 032011 cohen (1998，p. 7)将这个数称为“孪生素数和 goldbach 常数” ，并指出，在 x 趋于无穷大时，具有 p < = x 的孪生素数对(p，p + 2)的数目趋于2 * c _ 2 * x/log (x) ^ 2。阿图尔・贾辛斯基，2021年2月1日

白新岭 · 发表于 2021-7-8 18:01

REFERENCES
Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 11.

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, pp. 84-93.

Philippe Flajolet and Ilan Vardi, Zeta function Expansions of Classical constants, Feb 18 1996.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, ch. 22.20.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
参考 henri cohen，数论，第二卷: 分析与现代工具，第二卷。240，springer，2007; 参见第208-209页。R. crandall and c. pomerance，prime numbers: a computational perspective，springer，ny，2001; see p. 11.数学常数，数学百科全书及其应用，第卷。94，cambridge university press，2003，pp. 84-93.Philippe flajolet 和 ilan vardi，zeta 函数展开的经典常数，1996年2月18日。G. h. hardy 和 e.m. wright，《数论导论》，第五版，牛津大学出版社，1979年，第五版。22.20.N. j. a. sloane and simon plouffe，the encyclopedia of integer sequences，academic press，1995(includes this sequence).

白新岭 · 发表于 2021-7-8 18:04

LINKS
Harry J. Smith, Table of n, a(n) for n = 0..1001

Folkmar Bornemann, PRIMES Is in P: Breakthrough for "Everyman", Notices Amer. Math. Soc., 50(5) (May 2003), p. 549.

C. K. Caldwell, The Prime Glossary, twin prime constant

Henri Cohen, High-precision computation of Hardy-Littlewood constants, (1998).

Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]

Steven R. Finch, Mathematical Constants, Errata and Addenda, arXiv:2001.00578 [math.HO], 2020, Sec. 2.1.

Philippe Flajolet and Ilan Vardi, Zeta function expansions of some classical constants.

Daniel A. Goldston, Timothy Ngotiaoco and Julian Ziegler Hunts, The tail of the singular series for the prime pair and Goldbach problems, Functiones et Approximatio Commentarii Mathematici, Vol. 56, No. 1 (2017), pp. 117-141; arXiv preprint, arXiv:1409.2151 [math.NT], 2014.

R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2011, constant T_1^(2).

G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]

G. Niklasch, Twin primes constant.

Simon Plouffe, The twin primes constant.

Simon Plouffe, Plouffe's Inverter, The twin primes constant.

Pascal Sebah (pascal_sebah(AT)ds-fr.com), Numbers, constants and computation (gives 5000 digits).

Eric Weisstein's World of Mathematics, Twin Primes Constant.

Eric Weisstein's World of Mathematics, Twin Prime Conjecture.

Eric Weisstein's World of Mathematics, k-Tuple Conjecture.

Eric Weisstein's World of Mathematics, Prime Constellation.

John W. Wrench, Jr., Evaluation of Artin's constant and the twin-prime constant, Math. Comp., 15 (1961), 396-398.
链接 harry j. smith n 的表 a (n)表 n = 0。Folkmar bornemann 1001”普通人”的突破口，注意到了 amer。数学。50(5)(may 2003) ，p. 549. c.K. caldwell，首要词汇，孪生首要 constanthenri cohen，hardy-littlewood 常数的高精度计算，(1998)。亨利 · 科恩，哈迪-利特伍德常数的高精度计算。史蒂文 · r · 芬奇，数学常量，勘误表及附录，arxiv: 2001.00578[ math.ho ] ，2020，第2.1节。Philippe flajolet and ilan vardi，zeta 函数展开的一些古典常数 daniel a. goldston，timothy gotaoco and julian ziegler hunts，the tail of the singular series for the prime pair and goldbach problems，functiones et approximation commentarii，vol。56，no. 1(2017) ，pp. 117-141; arxiv preprint，arxiv: 1409.2151[ math.nt ] ，2014.R.j. mathar，hardy-littlewood 常数嵌入到所有正整数的无限乘积中，arxiv: 0903.2514[ math.nt ] ，2009-2011，constant t _ 1 ^ (2)。G. niklasch，一些数字理论常数: 1000位数值。G. niklasch，twin primes constant.simon plouffe，the twin primes constant.simon plouffe，plouffe，plouffe’s transversion，the twin primes constant.pascal sebah (at) ds-fr. com) ，数字，常数和计算(给出5000位数)。Eric weisstein 的数学世界，孪生素数恒等式 eric weisstein 的数学世界，孪生素数猜想 eric weisstein 的数学世界，k-tuple 猜想 eric weisstein 的数学世界，素数恒等式 john w. wrench，jr. ，对 artin 常数和孪生素数恒等式的评估，数学。15(1961) ，396-398.

白新岭 · 发表于 2021-7-8 18:05

FORMULA
Equals prod(k>=2, (zeta(k)*(1-1/2^k))^(-sum(d/k, mu(d)*2^(k/d))/k)). - Benoit Cloitre, Aug 06 2003

Equals 1/A167864. - Jonathan Sondow, Nov 18 2009

EXAMPLE
0.6601618158468695739278121100145557784326233602847334133194484233354056423...

MATHEMATICA
s[n_] := (1/n)*N[ Sum[ MoebiusMu[d]*2^(n/d), {d, Divisors[n]}], 160]; C2 = (175/256)*Product[ (Zeta[n]*(1 - 2^(-n))*(1 - 3^(-n))*(1 - 5^(-n))*(1 - 7^(-n)))^(-s[n]), {n, 2, 160}]; RealDigits[C2][[1]][[1 ;; 105]] (* Jean-François Alcover, Oct 15 2012, after PARI *)

digits = 105; f[n_] := -2*(2^n-1)/(n+1); C2 = Exp[NSum[f[n]*(PrimeZetaP[n+1] - 1/2^(n+1)), {n, 1, Infinity}, NSumTerms -> 5 digits, WorkingPrecision -> 5 digits]]; RealDigits[C2, 10, digits][[1]] (* Jean-François Alcover, Apr 16 2016, updated Apr 24 2018 *)
公式 = prod (k > = 2，(zeta (k) * (1-1/2 ^ k)) ^ (- sum (d/k，mu (d) * 2 ^ (k/d))/k))。- benoit cloitre 2003年8月6日等于1/a167864。- jonathan sondow 2009年11月18日例0.6601618158468695739278121001455778432623360284733413319448423354056423..。数学[ n _ ] : = (1/n) * n [ sum [ moebiusmu [ d ] * 2 ^ (n/d) ，{ d，divisors [ n ]} ，160] ; c2 = (175/256) * 乘积[(zeta [ n ] * (1-(zeta [ n ] * (1-2 ^ (- n)) * (1-3 ^ (- n))) * (1-5 ^ (- n))) * (1-7(- n))](1-2 ^ (- n)) * (1-3 ^ (- n)) * (1-5 ^ (- n)) * (1-7 ^ (- n))) ^ (- s [ n ]) ，{ n，2,160}] ;实数[ c2][[1][1; 105](* jean-françois alcover，oct 152012，after pari *)数字 = 105; f [ n _ ] : =-2 * (2 ^ n-1)/(n + 1) ;C2 = exp [ nsum [ n ] * (primezetap [ n + 1]-1/2 ^ (n + 1)) ，{ n，1，infinity } ，nsumterms-> 5位数字，计算精度-> 5位数字] ; realdigits [ c2,10，数字][1](* jean-françois alcover，162016，updated apr 242018 *)(* jean-françois alcover，apr 162016，updated apr 242018 *)

白新岭 · 发表于 2021-7-8 18:06

PROG
(PARI) \p1000; 175/256*prod(k=2, 500, (zeta(k)*(1-1/2^k)*(1-1/3^k)*(1-1/5^k)*(1-1/7^k))^(-sumdiv(k, d, moebius(d)*2^(k/d))/k))

(PARI) prodeulerrat(1-1/(p-1)^2, 1, 3) \\ Amiram Eldar, Mar 12 2021

CROSSREFS
Cf. A065645 (continued fraction), A065646 (denominators of convergents to twin prime constant), A065647 (numerators of convergents to twin prime constant), A062270, A062271, A114907, A065418 (C_3), A167864.

Sequence in context: A334710 A283203 A155742 * A281056 A273989 A197013

Adjacent sequences: A005594 A005595 A005596 * A005598 A005599 A005600

KEYWORD
cons,nonn,nice

AUTHOR
N. J. A. Sloane

EXTENSIONS
More terms from Vladeta Jovovic, Nov 08 2001

Commented and edited by Daniel Forgues, Jul 28 2009, Aug 04 2009, Aug 12 2009

PARI code removed by D. S. McNeil, Dec 26 2010

STATUS
approved
Prog (pari) p1000;175/256 * prod (k = 2,500，(zeta (k) * (1-1/2 ^ k) * (1-1/3 ^ k) * (1-1/5 ^ k) * (1-1/7 ^ k))(zeta (k) * (1-1/2 ^ k) * (1-1/3 ^ k) * (1-1/5 ^ k) * (1-1/7 ^ k)) ^ (- sumdiv (k，d，moebius (d) * 2 ^ (k/d)/k))(- sumdiv (k，d，moebius (d) * 2 ^ (k/d)/k))(pari) prodeulerrat (1-1/(p-1) ^ 2,1,3) amiram eldar，mar 122021/crossrefs cf.A065645(连分数) ，a065646(convergents to twin prime constant 的分母) ，a065647(convergents to twin prime constant 的分子) ，a062270，a062271，a114907，a065418(c _ 3)(c _ 3) ,167864.sequence in context: a334710 a283203 a155742 * a281056 a273989 a197013寸相邻序列: a005594 a005595 a005596 * a005598 a005599 a0056000cons，nonn，niceauthor n。J. a. sloaneextensions 更多术语来自 vladita jovovic 2001年11月8日 daniel forgues 评论和编辑2009年7月28日，2009年8月4日，2009年8月12日被 d. s. mcneil 删除的 pari 代码，2010年12月26日批准

白新岭 · 发表于 2021-7-8 18:09

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