Review Article Volume 2 Issue 6
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
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.
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,⋯k−1.2|ni,i=1,⋯k−1.
J2(a))=∏P(P−1−χ(P)),J2(a))=∏P(P−1−χ(P)), (2)
where co=∏PP,χ(P)co=∏PP,χ(P) is the number of solutions of congruence
∏k−1i=1(q+ni)≡0 (mod P), q=1,⋯,p−1∏k−1i=1(q+ni)≡0(modP),q=1,⋯,p−1 (3)
which is true.
If χ(P)<P−1χ(P)<P−1 then J2(ω)≠0J2(ω)≠0 . There exist infinitely many primes PP such that each of P+niP+ni is prime. If χ(P)=P−1χ(P)=P−1 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)=|{P≤N:P+ni=prime}|−J2(co)cok−1N∅k(co)logkN=C(k)NlogkNπk(N,2)=|{P≤N:P+ni=prime}|−J2(co)cok−1N∅k(co)logkN=C(k)NlogkN (4)
∅(co)=∏P(P−1)∅(co)=∏P(P−1)
C(k)=∏P(1−1+χ(P)P)(1−1P)−kC(k)=∏P(1−1+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)=1 if P>2χ(20,)=χ(P)1=ifP>2 (6)
Substituting (6) into (2) we have
J2(a))=∏P≥3(P−2)≠0J2(a))=∏P≥3(P−2)≠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)=|{P≤N:P+2=prime}|−2∏P≥3(1−1(P−1)2)Nlog2Nπk(N,2)=|{P≤N:P+2=prime}|−2∏P≥3(1−1(P−1)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)=3 if P>3χ(20,)=χ(31,)=χ(P)3=ifP>3 . (11)
Substituting (11) into (2) we have
J2(ω)=∏P≥5(P−4)≠0J2(ω)=∏P≥5(P−4)≠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)=|{P≤N:P+n=prime}|−273∏P≥5P3(P−4)N(P−1)4log4Nπ4(N,2)=|{P≤N:P+n=prime}|−273∏P≥5P3(P−4)N(P−1)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)=4 if P>5χ(20,)=χ(31,)=χ(53,)=χ(P)4=ifP>5 (14)
Substituting (14) into (2) we have
J2(ω)=∏P≥7(P−5)≠0J2(ω)=∏P≥7(P−5)≠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)=|{P≤N:P+n=prime}|−154211∏P≥7(P−5)P4N(P−1)5log5Nπ5(N,2)=|{P≤N:P+n=prime}|−154211∏P≥7(P−5)P4N(P−1)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,⋯,k−1.
In 1923 Hardy et al.3 conjectured the asymptotic formula
πk(N,2)=|{P≤N:P+ni=prime}|−H(k)NlogkN , (19)
where
H(k)=∏P(1−V(P)P)(1−1_P) (20)
Is Hardy-Littlewood wrong singula series,
v(P) is the number of solutions of congruence
∏k−1i=1(q+ni)≡0(mod P) , 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)=∏PPP−1 (23)
Substituting (23) into (19) we have the asymptotic formula
π2(N,2)=|{P≤N:P+2=prime}|−∏pPNP−1log2N (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)=2 if P>2 (25)
Substituting (25) into (20) we have
H(3)=4∏P≥3P2(P−2)(P−1)3 (26)
Substituting (26) into (19) we have asymptotic formula
π3(N,2)=|{P≤N:P+2=prime, P+4=prim}|−4∏P≥3P2(P−2)N(P−1)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)=3 if P>3 (28)
Substituting (28) into (20) we have
H(4)=272∏P>3P3(P−3)(P−1)4 (29)
Substituting (29) into (19) we have asymptotic formula
π4(N,2)=|{P≤N:P+n=prime}|−272∏P>3P3(P−3)N(P−1)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)=4 if P>5 (31)
Substituting (31) into (20) we have
H(5)=15445∏P>5P4(P−4)(P−1)5 (32)
Substituting (32) into (19) we have asymptotic formula
π5(N,2)=|{P≤N:P+n=prime}|−15445∏P>5P4(P−4)N(P−1)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)=5 if P>5 (34)
Substituting (34) into (20) we have
H(6)=155213∏P>5(P−5)P5(P−1)6 (35)
Substituting (35) into (19) we have asymptotic formula
π6(N,2)=|{P≤N:P+n=prime}|−155213∏P>5(P−5)P5N(P−1)6log6N (36)
which is wrong see example 5.
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.
None.
Author declares that there is no conflict of interest.
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