On the sngular seres n the Jang prme k-tuple theorem Chun-Xuan Jang. O. Box 9, Bejng 1008,. R. Chna jcxuan@sna.com Abstract Usng Jang functon we prove Jang prme k -tuple theorem.we fnd true sngular seres. Usng the examples we prove the Hardy-Lttlewood prme k -tuple conjecture wth wrong sngular seres.. Jang prme k -tuple theorem wll replace the Hardy-Lttlewood prme k -tuple conjecture. (A) Jang prme k -tuple theorem wth true sngular seres[1, ]. We defne the prme k -tuple equaton p p, (1), n where n, 1, k1. we have Jang functon [1, ] J ( ) ( 1 ( )), () where, ( ) s the number of solutons of congruence whch s true. k 1 ( qn ) 0 (mod ), q 1,, p1. () 1 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 n s prme. J ( ) s a subset of Euler functon ( )[]. If J ( ) 0, then we have the best asymptotc formula of the number of prme [1, ] 1 (,) : ~ ( ) k J k n prme C( k) k k k () ( ) log log ( ) ( 1) 1
1 ( ) 1 Ck ( ) 1 1 s Jang true sngular seres. Example 1.Let k,,, twn prmes theorem. From () we have k () () 0, ( ) 1 f, (6) Substtutng (6) nto () we have J ( ) ( ) 0 (7) There exst nfntely many prmes such that () we have the best asymptotc formula s prme. Substtutng (7) nto 1 k (,) : prme ~ (1 ). log (8) Example.Let k,,,. From () we have From () we have () 0, () (9) J ( ) 0. (10) It has only a soluton,, 7. One of,, s always dvsble by.example s not admssble. Example.Let k,, n,where n,6,8. From () we have Substtutng (11) nto () we have () 0, () 1, ( ) f. (11) J ( ) ( ) 0, (1) There exst nfntely many prmes such that each of n s prme.example s admssble. Substtutng (1) nto () we have the best asymptotc formula 7 ( ) n prme (,) : ~ ( 1) log (1) Example. Let k,, n,where n,6,8,1.
From () we have Substtutng (1) nto () we have () 0, () 1, (), ( ) f (1) J ( ) ( ) 0 (1) 7 There exst nfntely many prmes such that each of n s prme. Example s admssble.substtutng (1) nto () we have the best asymptotc formula 1 ( ) (,) : n prme ~ 11 7 log (16) Example.Let k 6,, n,where n,6,8,1,1. From()and()wehave () 0, () 1, (), J () 0 (17) It has only a soluton, 7, 6 11, 8 1, 1 17, 1 19.Oneof n s always dvsble by. Example s not admssble. ( B ) The Hardy-Lttlewood prme k -tuple conjecture wth wrong sngular seres[-17]. Ths conjecture s generally beleved to be true, but has not been proved(odlyzko et al. Jumpng champons,exper.math.,8.107-118.1999). We defne the prme k -tuple equaton where n, 1,, k1., n (18) In 19 Hardy and Lttlewood conjectured the asymptotc formula where k(,) : n prme ~ H( k), (19) log k ( ) 1 Hk ( ) 1 1 s Hardy-Lttlewood wrong sngula seres, k (0) ( ) s the number of solutons of congruence whch s wrong. k 1 ( qn ) 0 (mod ), q 1,,. (1) 1
From (1) we have ( ) and Hk ( ) 0.Foranyprme k -tuple equaton there exst nfntely many prmes such that each of n s prme, whch s false. Conjecture 1.Let k,,, twn prmes theorem From (1) we have ( ) 1 () Substtutng () nto (0) we have H () () 1 Substtutng () nto (19) we have the asymptotc formula (,) : prme ~ 1log () whch s wrong see example 1.They do not get twn prmes formula(8). Conjecture.Let k,,,. From (1) we have Substtutng () nto (0) we have () 1, ( ) f () H () ( ) ( 1) (6) Substtutng (6) nto (19) we have asymptotc formula ( ) (,) : prme, prm ~ whch s wrong see example. Conjecture.Let k,, n,where n,6,8. log (7) From (1) we have () 1, (), ( ) f (8) Substtutng (8) nto (0) we have H () 7 ( ) (9) Substtutng (9) nto (19) we have asymptotc formula 7 ( ) (,) : n prme ~ log (0)
Whch s wrong see example. Conjecture.Let k,, n,where n,6,8,1 From (1) we have () 1, (), (), ( ) f (1) Substtutng (1) nto (0) we have H () ( 1) 1 ( ) () Substtutng () nto (19) we have asymptotc formula 1 ( ) (,) : n prme ~ log Whch s wrong see example. Conjecture.Let k 6,, n,where n,6,8,1,1. () From (1) we have () 1, (), (), ( ) f () Substtutng () nto (0) we have 1 ( ) H (6) 1 6 ( 1) () Substtutng () nto (19) we have asymptotc formula 1 ( ) 6(,) : n prme ~ 1 6 log 6 (6) whch s wrong see example. Concluson. The Jang prme k-tuple theorem has true sngular seres.the Hardy-Lttlewood prme k -tuple conjecture has wrong sngular seres.. The tool of addtve prme number theory s bascally the Hardy-Lttlewood wrong prme k-tuple conjecture whch are wrong[-17]. Usng Jang true sngula seres we prove almost all prme theorems. Jang prme k -tuple theorem wll replace Hardy-Lttlewood prme k -tuple Conjecture. There cannot be really modern prme theory wthout Jang functon. References
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