On the sngular seres n the Jang prme -tuple theorem Chun-Xuan Jang. O. Box 94, Bejng 10084,. R. Chna jcxuan@sna.com Abstract Usng Jang functon we prove Jang prme -tuple theorem.we fnd true sngular seres. Usng the examples we prove the Hardy-Lttlewood prme -tuple conjecture wth wrong sngular seres.. Jang prme -tuple theorem wll replace the Hardy-Lttlewood prme -tuple conjecture. (A) Jang prme -tuple theorem wth true sngular seres[1, ]. We defne the prme -tuple equaton p p, (1), n where n, 1, 1. we have Jang functon [1, ] J ( ) ( 1 ( )), () where, ( ) s the number of solutons of congruence 1 ( qn ) 0 (mod ), q 1,, p1. () 1 whch s true. If ( ) 1 then J ( ) 0. There exst nfntely many prmes such that each of n s prme. If ( ) 1 then J ( ) 0. There exst fntely many prmes such that each of s prme. J ( ) s a subset of Euler functon n ( )[]. If J ( ) 0, then we have the best asymptotc formula of the number of prme [1, ] 1 (,) : ~ ( ) J n prme C( ) ( ) log log (4) ( ) ( 1) 1
1 ( ) 1 C ( ) 1 1 s Jang true sngular seres. Example 1. Let,,, twn prmes theorem. From () we have () Substtutng (6) nto () we have There exst nfntely many prmes () 0, ( ) 1 f, (6) J ( ) ( ) 0 (7) (4) we have the best asymptotc formula such that s prme. Substtutng (7) nto 1 (,) : prme ~ (1 ). ( 1) log (8) Example. Let,,, 4. From () we have From () we have () 0, () (9) J ( ) 0. (10) It has only a soluton,, 4 7. One of,, 4 s always dvsble by. Example. Let 4,, n, where n,6,8. From () we have Substtutng (11) nto () we have () 0, () 1, ( ) f. (11) J ( ) ( 4) 0, (1) There exst nfntely many prmes such that each of n s prme. Substtutng (1) nto (4) we have the best asymptotc formula 7 ( 4) 4 n prme (,) : ~ ( 1) log (1) Example 4. Let,, n, where n,6,8,1. From () we have
Substtutng (14) nto () we have () 0, () 1, (), ( ) 4 f (14) J ( ) ( ) 0 (1) 7 There exst nfntely many prmes (1) nto (4) we have the best asymptotc formula such that each of n s prme. Substtutng 1 ( ) (,) : n prme ~ 11 7 ( 1) log (16) Example. Let 6,, n, where n,6,8,1,14. From () and () we have () 0, () 1, () 4, J () 0 (17) It has only a soluton, 7, 6 11, 8 1, 1 17, 14 19. One of n s always dvsble by. (B)The Hardy-Lttlewood prme -tuple conjecture wth wrong sngular seres[-14]. Ths conjecture s generally beleved to be true, but has not been proved(odlyzo et al.1999). We defne the prme -tuple equaton where n, 1,, 1., n (18) In 19 Hardy and Lttlewood conjectured the asymptotc formula where (,) : n prme ~ H( ), (19) log ( ) 1 H ( ) 1 1 s Hardy-Lttlewood wrong sngula seres, (0) ( ) s the number of solutons of congruence whch s wrong. 1 ( qn ) 0 (mod ), q1,,. (1) 1 From (1) we have ( ) and H ( ) 0. For any prme -tuple equaton there
exst nfntely many prmes Conjecture 1. Let From (1) we have such that each of,,, twn prmes theorem n s prme, whch s false. ( ) 1 () Substtutng () nto (0) we have H () () 1 Substtutng () nto (19) we have the asymptotc formula (,) : prme ~ 1log (4) whch s wrong see example 1. Conjecture. Let,,, 4. From (1) we have () 1, ( ) f () Substtutng () nto (0) we have H () 4 ( ) ( 1) (6) Substtutng (6) nto (19) we have asymptotc formula ~ 4 ( ) (,) : prme, 4 prm whch s wrong see example. Conjecture. Let 4,, n, where n, 6,8. ( 1) log (7) From (1) we have () 1, (), ( ) f (8) Substtutng (8) nto (0) we have H (4) 7 ( ) 4 ( 1) (9) Substtutng (9) nto (19) we have asymptotc formula 7 ( ) 4(,) : n prme ~ ( 1) log Whch s wrong see example. (0) 4
Conjecture 4. Let,, n, where n,6,8,1 From (1) we have () 1, (), (), ( ) 4 f (1) Substtutng (1) nto (0) we have H () 1 ( 4) 4 ( 1) () Substtutng () nto (19) we have asymptotc formula 1 ( 4) n prme (,) : ~ () 4 ( 1) log Whch s wrong see example 4. Conjecture. Let 6,, n, where n,6,8,1,14. From (1) we have () 1, (), () 4, ( ) f (4) Substtutng (4) nto (0) we have 1 ( ) H (6) 1 6 ( 1) () Substtutng () nto (19) we have asymptotc formula 1 ( ) 6(,) : n prme ~ 1 ( 1) 6 log 6 (6) whch s wrong see example. Concluson. The Jang prme -tuple theorem has true sngular seres.the Hardy-Lttlewood prme -tuple conjecture has wrong sngular seres.. The tool of addtve prme number theory s bascally the Hardy-Lttlewood wrong prme -tuple conjecture whch are wrong[-14]. Usng Jang true sngula seres we prove almost all prme theorems. Jang prme -tuple theorem wll replace Hardy-Lttlewood prme -tuple Conjecture. There cannot be really modern prme theory wthout Jang functon. References
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