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Physics & Astronomy International Journal

Review Article Volume 2 Issue 6

On the singular series in the Jiang prime k-tuple theorem

Chun Xuan Jiang

China Aerospace Science and Technology Corporation, China

Correspondence: Chun Xuan Jiang, China Aerospace Science and Technology Corporation, P. O. Box 142-206, Beijing 100854, P. R. China

Received: September 18, 2018 | Published: November 16, 2018

Citation: Jiang CX. On the singular series in the Jiang prime k-tuple theorem. Phys Astron Int J. 2018;2(6):514-517. DOI: 10.15406/paij.2018.02.00134

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Abstract

Using Jiang function we prove Jiang prime k-tuple theorem. We find true singular series. Using the examples we prove the Hardy-Littlewood prime k-tuple conjecture with wrong singular series. Jiang prime k-tuple theorem will replace the Hardy-Littlewood prime k-tuple conjecture.

Theorems

Jiang prime k-tuple theorem with true singular series.1,2

We define the prime k-tuple equation

p,p+ni,p,p+ni,   (1)

where 2|ni,i=1,k1.2|ni,i=1,k1.

we have Jiang function1,2

J2(a))=P(P1χ(P)),J2(a))=P(P1χ(P)),   (2)

where co=PP,χ(P)co=PP,χ(P)  is the number of solutions of congruence

k1i=1(q+ni)0(modP),q=1,,p1k1i=1(q+ni)0(modP),q=1,,p1   (3)

which is true.

If χ(P)<P1χ(P)<P1  then J2(ω)0J2(ω)0 . There exist infinitely many primes PP  such that each of P+niP+ni  is prime. If χ(P)=P1χ(P)=P1  then J2(ω)=0J2(ω)0= . There exist finitely many primes PP  such that each of P+niP+ni  is prime. J2(ω)J2(ω)  is a subset of Euler function (co)(co) .2

If J2(ω)0J2(ω)0 , then we have the best asymptotic formula of the number of prime PP .1,2

πk(N,2)=|{PN:P+ni=prime}|J2(co)cok1Nk(co)logkN=C(k)NlogkNπk(N,2)=|{PN:P+ni=prime}|J2(co)cok1Nk(co)logkN=C(k)NlogkN   (4)

(co)=P(P1)(co)=P(P1)

C(k)=P(11+χ(P)P)(11P)kC(k)=P(11+χ(P)P)1(1P)k   (5)

is Jiang true singular series.

Example 1

Let k=2,P,P+2k=2,P,P+2 , twin primes theorem.

From (3) we have

χ(2)=0,χ(P)=1ifP>2χ(20,)=χ(P)1=ifP>2   (6)

Substituting (6) into (2) we have

J2(a))=P3(P2)0J2(a))=P3(P2)0   (7)

There exist infinitely many primes PP  such that P+2P+2  is prime. Substituting (7) into (4) we have the best asymptotic formula

πk(N,2)=|{PN:P+2=prime}|2P3(11(P1)2)Nlog2Nπk(N,2)=|{PN:P+2=prime}|2P3(11(P1)2)Nlog2N   (8)

Example 2

Let k=3,P,P+2,P+4.k=3,P,P+2,P+4.

From (3) we have

χ(2)=0,χ(3)=2χ(20,)=χ(32)=   (9)

From (2) we have

J2(ω)=0J2(ω)0=   (10)

It has only a solution P=3,P+2=5,P+4=7P=3,P+25,=P+47= . One of P,P+2,P+4P,P+2,P+4  is always divisible by 3. Example 2 is not admissible.

Example 3

Let k=4,P,P+nk=4,P,P+n , where n=2,6,8.n=2,6,8.

From (3) we have

χ(2)=0,χ(3)=1,χ(P)=3ifP>3χ(20,)=χ(31,)=χ(P)3=ifP>3 .  (11)

Substituting (11) into (2) we have

J2(ω)=P5(P4)0J2(ω)=P5(P4)0 ,  (12)

There exist infinitely many primes PP  such that each of P+nP+n  is prime. Example 3 is admissible.

Substituting (12) into (4) we have the best asymptotic formula

π4(N,2)=|{PN:P+n=prime}|273P5P3(P4)N(P1)4log4Nπ4(N,2)=|{PN:P+n=prime}|273P5P3(P4)N(P1)4log4N   (13)

Example 4

Let k=5,P,P+nk=5,P,P+n , where n=2,6,8,12.n=2,6,8,12.

From (3) we have

χ(2)=0,χ(3)=1,χ(5)=3,χ(P)=4ifP>5χ(20,)=χ(31,)=χ(53,)=χ(P)4=ifP>5   (14)

Substituting (14) into (2) we have

J2(ω)=P7(P5)0J2(ω)=P7(P5)0   (15)

There exist infinitely many primes PP  such that each of P+nP+n  is prime. Example 4 is admissible. Substituting (15) into (4) we have the best asymptotic formula

π5(N,2)=|{PN:P+n=prime}|154211P7(P5)P4N(P1)5log5Nπ5(N,2)=|{PN:P+n=prime}|154211P7(P5)P4N(P1)5log5N   (16)

Example 5

Let k=6,P,P+nk=6,P,P+n , where n=2,6,8,12,14.n=2,6,8,12,14.

From (3) and (2) we have

χ(2)=0,χ(3)=1,χ(5)=4,J2(5)=0χ(20,)=χ(31,)=χ(54,)=J2(50)=   (17)

It has only aa  solution P=5,P+2=7,P+6=11,P+8=13,P+12=17,P+14=19P=5,P+27,=P+611,=P+813,=P+1217,=P+1419= . One of P+n  is always divisible by 5. Example 5 is not admissible.

The Hardy-Littlewood prime k-tuple conjecture with wrong singular series.3−17

This conjecture is generally believed to be true, but has not been proved.18

We define the prime k-tuple equation

P,P+ni   (18)

where 2|ni,i=1,,k1.

In 1923 Hardy et al.3 conjectured the asymptotic formula

πk(N,2)=|{PN:P+ni=prime}|H(k)NlogkN ,  (19)

where

H(k)=P(1V(P)P)(11_P)   (20)

Is Hardy-Littlewood wrong singula series,

v(P)  is the number of solutions of congruence

k1i=1(q+ni)0(modP) , q=1,,P .  (21)

which is wrong.

From (21) we have v(P)<P  and H(k)0 . For any prime k-tuple equation there exist infinitely many primes P  such that each of P+ni  is prime, which is false.

Conjecture 1

Let k=2,P,P+2 , twin primes theorem

From (21) we have

v(P)=1   (22)

Substituting (22) into (20) we have

H(2)=PPP1   (23)

Substituting (23) into (19) we have the asymptotic formula

π2(N,2)=|{PN:P+2=prime}|pPNP1log2N   (24)

which is wrong see example l. They do not get twin primes formula (8).

Conjecture 2

Let k=3,P,P+2,P+4.

From (21) we have

v(2)=1,v(P)=2ifP>2   (25)

Substituting (25) into (20) we have

H(3)=4P3P2(P2)(P1)3   (26)

Substituting (26) into (19) we have asymptotic formula

π3(N,2)=|{PN:P+2=prime,P+4=prim}|4P3P2(P2)N(P1)3log3N   (27)

which is wrong see example 2.

Conjecture 3

Let k=4,P,P+n , where n=2,6,8.

From (21) we have

v(2)=1,v(3)=2,v(P)=3ifP>3   (28)

Substituting (28) into (20) we have

H(4)=272P>3P3(P3)(P1)4   (29)

Substituting (29) into (19) we have asymptotic formula

π4(N,2)=|{PN:P+n=prime}|272P>3P3(P3)N(P1)4log4N   (30)

Which is wrong see example 3.

Conjecture 4

Let k=5,P,P+n , where n=2,6,8,12

From (21) we have

v(2)=1,v(3)=2,v(5)=3,v(P)=4ifP>5   (31)

Substituting (31) into (20) we have

H(5)=15445P>5P4(P4)(P1)5   (32)

Substituting (32) into (19) we have asymptotic formula

π5(N,2)=|{PN:P+n=prime}|15445P>5P4(P4)N(P1)5log5N   (33)

Which is wrong see example 4.

Conjecture 5

Let k=6,P,P+n , where n=2,6,8,12,14.

From (21) we have

v(2)=1,v(3)=2,v(5)=4,v(P)=5ifP>5   (34)

Substituting (34) into (20) we have

H(6)=155213P>5(P5)P5(P1)6   (35)

Substituting (35) into (19) we have asymptotic formula

π6(N,2)=|{PN:P+n=prime}|155213P>5(P5)P5N(P1)6log6N   (36)

which is wrong see example 5.

Conclusion

The Jiang prime k-tuple theorem has true singular series.The Hardy-Littlewood prime -tuple conjecture has wrong singular series. The tool of additive prime number theory is basically the Hardy-Littlewood wrong prime k-tuple conjecture which are wrong.3−17 Using Jiang true singula series we prove almost all prime theorems. Jiang prime k-tuple theorem will replace Hardy-Littlewood prime k-tuple Conjecture. There cannot be really modern prime theory without Jiang function.

Acknowledgments

None.

Conflicts of interest

Author declares that there is no conflict of interest.

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