素数分布中蒋函数 蒋春暄. 应该是 ν (log N ) k, 不是 ν 2 (log N ) k 他们 66 页论文没有直接讨论素

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1 素数分布中蒋函数 蒋春暄 777 年最伟大数学家 Euler 说 : 数学家还没有发现素数序列中的一些规则 我们有理由相信它是一个人类智慧尚未洞悉的奥秘 0 世纪最伟大数学家 Erdös 说 : 至少还需要 00 万年, 我们才能真正理解素数 说明素数研究多么困难! 多么复杂! 但是我的兴趣就是要研究没有人研究的问题 用我的思路 我的方法进行研究 不管这个问题多么困难 这篇论文是把我过去对素数研究作一个总结, 使人们更容易理解 我发现只有素数定理才算一个真正定理 : 即欧几里德证明了有无限多个素数, 并找到计算低于 N 素数个数公式 N/logN, 其它素数定理都是猜想, 因为只获得上限和下限公式并没有证明它有无限多素数解 用蒋函数我证明了素数分布中几乎所有问题 素数问题是有规则的, 不是随机的 本文只有一个定理 孪生素数和哥德巴赫猜想只能作为两个特例 这个定理包括素数分布所有问题 素数研究最近最大成果是格林和陶哲轩证明存在任意长的素数等差数列 陶哲轩因此获得 006 年国际数学家大会菲尔茨奖 王元对陶哲轩评价 : 我不敢想象天下会有这样伟大的成就 陶哲轩文章内容就是本文 Example8 格林 陶哲轩论文是错误的 : 他们没有证明 < 素数等差数列 > 无限多素数解 他们没有找计算素数个数的公式, 他们是抄前人的公式, 计算个数概念不清 请看文献 [0], 公式 应该是 ν (log N ), 不是 ν (log N ) 他们 66 页论文没有直接讨论素 N N 数等差数列, 他们根本不懂素数, 普林斯顿 数学年刊 发表这样一篇错误的论文, 说明这杂志编委不懂素数, 因为全世界数学家都不懂素数, 把这篇错误论文评为 006 年国际数学家的菲尔茨奖, 这就是当代素数研究水平 本文 Example 8 已给素数等差数列彻底解决

2 Jiag s fuctio J ( ) + ω i prime distributio Chu-Xua Jiag P. O. Box 94, Beijig 00854, P. R. Chia jiagchuxua@vip.sohu.com Dedicated to the 0-th aiversary of hadroic mechaics We defie that prime equatios Abstract f ( P, L, P ), L, f ( P, L P ) (5) are polyomials (with iteger coefficiets) irreducible over itegers, where P, L, P are all prime. If Jiag s fuctio J ( ) 0 + ω = the (5)has fiite prime solutios. If J ( ) 0 + ω the there are ifiitely may primes P, L, P such that f L f are all prime. We obtai a uite prime formula i prime distributio, { L L } + ( N, + ) = P,, P N: f,, f are all prime J + ( ωω ) N ~ = (deg fi ) ( + o()). (8) + +! φ ( ω) log N i= Jiag s fuctio is accurate sieve fuctio. Usig Jiag s fuctio we prove about 600 prime theorems[6]. Jiag s fuctio provids proofs of the prime theorems which are simple eough to uderstad ad accurate eough to be useful.

3 Mathematicias have tried i vai to discover some order i the sequece of prime umbers but we have every reaso to believe that there are some mysteries which the huma mid will ever peetrate. Leohard Euler It will be aother millio years, at least, before we uderstad the primes. Paul Erdös Support that Euler totiet fuctio φ( ω) = Π( P ) = as ω, () P where ω =Π P is called primorial. P Support that ( ω, h i ) =, where i =, L, φ( ω). We have prime equatios P = ω+,, P φ ( ω ) = ω+ h φ( ω) L () where = 0,,, L. ()is called ifiitely may prime equatios (IMPE). Every equatio has ifiitely may prime solutios. We have ( N) h = ( o()). i = +, () φω ( ) Pi N Pi hi(mod ω) where h i deotes the umber of primes Pi N i P i = ω+ h i = 0,,, L, ( N ) the umber of primes less tha or equal to N. We replace set of prime umbers by IMPE. () is the fudametal tool for provig the prime theorems i prime distributio. Let ω = 0 ad φ (0) = 8. From () we have eight prime equatios P = 0+, P = 0+ 7, P = 0+, P4 = 0+, P5 = 0+ 7, P6 = 0+ 9, P7 = 0+, P8 = 0+ 9, = 0,,, L (4) Every equatio has ifiitely may prime solutios. THEOREM. We defie that prime equatios f ( P, L, P ), L, f ( P, L, P) (5)

4 are polyomials (with iteger coefficiets) irreducible over itegers, where P, L, P are all prime. There exist ifiitely may such that each f is prime. PROOF. Firstly, we have Jiag s fuctio [-] J + tuplets of P,, L P ( ω) =Π[( P ) χ( P)], (6) P where χ ( P) is called sieve costat ad deotes the umber of solutios for the followig cogruece Π f ( q, L, q ) 0 (mod P), (7) i= i where q =, L, P, L, q =, L, P. J ( ) + ω deotes the umber of tuplets of P, L, P prime equatios for which f( P, L, P), L, f( P, L, P) are prime equatio. If J ( ) 0 + ω = the (5) has fiite prime solutios. If J ( ) 0 + ω usig χ ( P) we sift out from () prime equatios that ca ot be represeted P, L, P, the residual prime equatios of () are P, L, P prime equatios for which f( P, L, P ), L, f ( P, L, P) are all prime equatio, hece we prove that there exist ifiitely may tuplets of primes P, L, P for which f ( P, L, P) are all prime. Secodly, we have the best asymptotic formula [,,4,6] { L L } + ( N, + ) = P,, P N: f,, fare all prime f P L P (,, ), L, =Π J ( ) N (deg f ) + ωω i ( ()). i= + +! φ ( ω) log N +o (8) (8)is called a uite prime formula i prime distributio. Let =, = 0, J ( ω) φω ( ) =. From (8) we have prime umber theorem N ( N, ) = { P N : Pis prime } = ( + o()).. (9) log N Number theorists believe that there are ifiitely may twi primes, but they do ot have rigorous proof of this old cojecture by ay method. All prime 4

5 theorems are cojectures except the prime umber theorm, because they do ot prove that prime equatios have ifiitely may prime solutios. We prove the followig cojectures by this theorem. Example. Twi primes PP+, (00BC). From (6) ad (7) we have Jiag s fuctio J ( ω) = Π( P ) 0 P Sice J ( ) 0 ω i () exist ifiitely may P prime equatios for which P + is prime equatio, hece we prove that there are ifiitely may primes P for which P + is prome. Let ω = 0 ad J (0) =. From (4) we have three P prime equatis P = 0+, P = 0+ 7, P = J ( ω) ω N ( N,) = { P N : P + prime } = ( o()) φ ( ω) log N + N = Π ( o()). P + ( P ) log N I 996 we proved twi primes cojecture [] Remar. J ( ) ω deotes the umber of P prime equatios, ω N ( + o()) the umber of solutios of primes for every P prime φ ( ω) log N equatio. Example. Eve Goldbach s cojecture N = P+ P. Every eve umber N 6 is the sum of two primes. From (6) ad (7) we have Jiag s fuctio 5

6 P J( ω) = Π( P ) Π 0. P P N P Sice J ( ) 0 ω as N i () exist ifiitely may P prime equatios for which N P is prime equatio, hece we prove that every eve umber N 6 is the sum of two primes. J( ω) ω N ( N, ) = { P N, N P prime } = ( o()). φ ( ω) log N + P N = Π ( o()) P Π +. ( P ) P N P log N I 996 we proved eve Goldbach s cojecture [] Example. Prime equatios PP, +, P+6. From (6) ad (7) we have Jiag s fuctio J ( ω) = Π( P ) 0, 5 P J ( ) ω is deotes the umber of P prime equatios for which P + ad P + 6 are all prime equatio. Sice J ( ) 0 ω i () exist ifiitely may P prime equatios such that prove that there are ifiitely may primes all prime. P + ad P + 6 are all prime equatio, hece we P such that P + ad P + 6 are Let ω = 0, J(0) =. From (4) we have two P prime equatios P = 0+, P = J( ωω ) N ( N,) = { P N : P +, P + 6 prime } = ( o()) φ ( ω) log N +. Example 4. Odd Goldbach s cojecture N = P+ P + P. Every odd umber N 9 is the sum of three primes. From (6) ad (7) we have Jiag s fuctio 6

7 J( ω) =Π( P P+ ) ) Π 0 P P N. P P+ Sice J ( ) 0 ω as N i () exist ifiitely may pairs of P ad P prime equatios for which N P P is prime equatio, hece we prove that every odd umber N 9 is the sum of three primes. J( ωω ) N ( N,) = { P, P N: N P P prime } = ( o()) φ ( ω) log N +. N =Π + ( ()) P Π o +. ( P ) P N P P+ log N Example 5. Prime equatio P = PP +. From (6) ad (7) we have Jiag s fuctio P ( ) J ( ω) = Π P P+ 0 J ( ) ω deotes the umber of pairs of P ad P prime equatios for which P is prime equatio. Sice J ( ) 0 ω i () exist ifiitely may pairs of ad prime equatios for which P is prime equatio, hece we prove that P there are ifiitely may pairs of primes P ad P for which P is prime. J( ωω ) N ( N, ) = { P, P N: PP + prime } = ( o()). 4 φ ( ω) log N + Note. deg ( PP ) =. Example 6 []. Prime equatio P = P + P. From (6) ad (7) we have Jiag s fuctio J P P ( ω) = Π ( ) χ( ) 0 P P where χ ( P) = ( P ) if P (modp ) ; χ ( P) = 0 if P / (modp ) ; 7

8 χ ( P) = P otherwise. Sice J ( ) 0 ω i () there are ifiitely may pairs of ad P prime equatios for which P P is prime equatio, hece we prove that there are ifiitely may pairs of primes P ad P for which P is prime. J( ωω ) N ( N,) = { P, P N: P + P prime } ~ ( o()). 6 φ ( ω) log N + 4 Example 7 []. Prime equatio P = P + ( P + ). From (6) ad (7) we have Jiag s fuctio J P P ( ω) = Π ( ) χ( ) 0 P where χ ( P) = ( P ) if P (mod 4) ; χ ( P) = ( P ) if P (mod 8) ; χ ( P) = 0 otherwise. Sice J ( ) 0 ω i () there are ifiitely may pairs of ad P prime equatios for which P P is prime equatio, hece we prove that there are ifiitely may pairs of primes P ad P for which P is prime. J( ωω ) N ( N,) = { P, P N: P prime } = ( o()). 8 φ ( ω) log N + Example 8 [4-9]. Arithmetic progressios cosistig oly of primes. We defie the arithmetic progressios of legth. P, P = P + d, P = P + d, L, P = P + ( ) d,( P, d) =. (0) { L } (,) = :, +,, + ( ) areallprime N P N P P d P d ( ) J ωω N = ( + o()).. φ ( ω) log N If J ( ω ) = 0 the (0) has fiite prime solutois. If J ( ) 0 ω the there are 8

9 ifiitely may primes P for which P, L, P are all prime. I [0] Cojecture. (Hardy-Littlewood cojecture o term APs) ad Theorem.(G.-Tao) are false. To elimiate d from (0) we have P = P P, P = ( j ) P ( j ) P, j. j From (6) ad (7) we have Jiag s fuctio J ( ω) = Π ( P ) Π( P )( P + ) 0 P< P Sice J ( ) 0 ω i () there are ifiitely may pairs of ad P prime P equatios for which P, L, P are all prime equatio, hece we prove that there are ifiitely may pairs of primes P ad P for which P, L, P are all prime. { } = ( N,) P, P N :( j ) P ( j ) P prime, j ( ) J ωω N = ( + o()) φ ( ω) log N P P ( P + ) N = Π Π ( + o()). P< ( P ) P ( P ) log N Example 9. It is a well-ow cojecture that oe of PP,, + P+ is always divisible by. To geeralize above to the primes, we prove the followig cojectures. Let. PP, + P, +, where ( + ). Frome (6) ad (7) we have divisible by.. PP, + P, +, L, P+ 4, where 5( + b), b=,. From (6) ad (7) we have be a square-free eve umber. J () = 0, hece oe of PP, + P, + is always J (5) = 0 4, hece oe of PP, + P, +, L, P+ 9

10 is always divisible by 5.. PP, + P, +, L, P+ 6, where 7( + b), b=,4. From (6) ad (7) we have is always divisible by PP, + P, +, L, P+, where ( + b), b=,4,5,9. From (6) ad (7) we have is always divisible by. 5. PP, + P, +, L, P+, where ( + b), b=,6,7,. From (6) ad (7) we have is always divisible by PP, + P, +, L, P+, where 7 ( + b), b=,5,6,7,0,,,4,5. From (6) ad (7) we have is always divisible by PP, + P, +, L, P+, where 9 ( + b), b= 4,5,6,9,6.7. From (6) ad (7) we have is always divisible by 9. Example 0. Let J (7) = 0 6, hece oe of PP, + P, +, L, P+ J () = 0, hece oe of PP, + P, +, L, P+ 0 J () = 0, hece oe of PP, + P, +, L, P+ J (7) = 0, hece oe of PP, + P, +, L, P+ 6 J (9) = 0, hece oe of PP, + P, +, L, P+ 8 be a eve umber.. PP, + i, i=,,5, L,+, From (6) ad (7) we have J ( ) 0 ω, hece we prove that there exist ifiitely i may primes P such that PP, + are all prime for ay.. PP, + i, i=, 4,6, L,. From (6) ad (7) we have J ( ) 0 ω, hece we prove that there exist ifiitely i may primes P such that PP, + are all prime for ay. Example. prime equatio P = P+ P 0

11 Frome (6) ad (7) we have Jiag s fuctio J P P ( ω) =Π( + ) 0 P Sice J ( ) 0 ω i () there are ifiitely may pairs of ad P prime equatios for which P P is prime equatio, hece we prove that there are ifiitely may pairs of primes P ad P for which P is prime. J( ωω ) N ( N,) = { P, P N: P prime } = ( o()). φ ( ω) log N + I the same way we ca prove = + which has the same Jiag s P P P fuctio. Jiag s fucito is accurate sieve fuctio. Usig it we ca prove ay irreducible prime equatios i prime distributio. There are ifiitely may twi primes but we do ot have rigorous proof of this old cojecture by ay method [0]. As strog as the umerical evidece may be, we still do ot eve ow whether there are ifiitely may pairs of twi primes []. All the prime theorems are cojectures except the prime umber theorem, because they do ot prove the simplest twi primes. They cojecture that the prime distributio is radomess [-5],because they do ot uderstad theory of prime umbers. Acolwdgemets The Author would lie to express his deepest appreciatio to M. N. Huxley, R. M. Satilli, L. Schadec ad G. Weiss for their helps ad supports. Refereces [] Chu-Xua Jiag, O the Yu-Goldbach prime theorem, Guagxi Scieces (Chiese) (996), 9-. [] Chu-Xua Jiag, Foudatios of Satilli s isoumber theory, Part I, Algebras Groups ad Geometries, 5(998), 5-9. [] ChuXua Jiag, Foudatios of Satilli s isoumber theory, Part II, Algebras Groups ad Geometries, 5(998), [4] Chu-Xua Jiag, Foudatios Satilli s isoumber theory, I: Foudametal ope problems i scieces at the ed of the milleium, T.

12 Gill, K. Liu ad E. Trell (Eds) Hadroic Press, USA, (999), [5] Chu-Xua Jiag, Proof of Schizel s hypothesis, Algebras Groups ad Geometries, 8(00), [6] Chu-Xua Jiag, Foudatios of Satilli s isomuber theory with applicatios to ew cryptograms, Fermat s theorem ad Goldbach s cojecture, Iter. Acad. Press, 00, MR004c: 00, [7] Chu-Xua Jiag,Prime theorem i Satilli s isoumber theory, 9(00), [8] Chu-Xua Jiag, Prime theorem i Satilli s isoumber theory (II), Algebras Groups ad Geometries, 0(00), [9] Chu-Xua Jiag, Disproofs of Riema s hypothesis, Algebras Groups ad Geometries, (005), Riema.pdf [0] Chu-Xua Jiag, Fiftee cosecutive itegers with exactly prime factors, Algebras Groups ad Geometries, (006), 9-4. [] Chu-Xua Jiag, The simplest proofs of both arbitrarily log arithmetic progressio of primes, preprit, 006. [] D. R. Heath-Brow, Primes represeted by x + y, Acta Math., 86 (00), -84. [] J. Friedlader ad H. Iwaiec, The polyomial x + y 4 captures its primes, A. Math., 48(998), [4] E. Szemerédi, O sets of itegers cotaiig o elemets i arithmetic progressios, Acta Arith., 7(975), [5] H. Fursteberg, Ergodic behavior of diagoal measures ad a theorem of Szemerédi o arithmetic progressios, J. Aalyse Math., (997), [6] W. T. Gowers, A ew proof of Szemerédi s theorem, GAFA, (00), [7] B. Kra, The Gree-Tao theorem o arithmetic progressios i the primes: A ergodic poit of view, Bull. Amer. Math. Soc., 4(006), -. [8] B. Gree ad T. Tao, The primes cotai arbitrarily log arithmetic progressios, A. Math., 67(08), [9] T. Tao, The dichotomy betwee structure ad radomess, arithmetic

13 progressios, ad the primes, I: proceedigs of the iteratioal cogress of mathematicias (Madrid. 006), Europ. Math. Soc. Vol , 007. [0] B. Gree, Log arithmetic progressios of primes, Clay Mathematics Proceedigs Vol. 7, 007, [] H. Iwaice ad E. Kowalsi, Aalytic umber theory, Amer. Math. Soc., Providece, RI, 004 [] R. Cradall ad C. Pomerace, Prime umbers a computatioal perpective, Sprig-Verlag, New Yor, 005. [] B. Gree, Geeralisig the Hardy-Littlewood method for primes, I: Proceedigs of the iteratioal cogress of mathematicias (Madrid. 006), Europ. Math. Soc., Vol. II, 7-99, 007. [4] K. Soudararaja, Small gaps betwee prime umbers: The wor of Goldsto-Pitz-Yildirim, Bull. Amer. Math. Soc., 44(007), -8. [5] A. Graville, Harald Cramér ad distributio of prime umbers, Scad. Actuar. J, 995()(995), -8.