aa r X i v : . [ m a t h . G M ] N ov Non-trivial zeros of Riemann’s Zeta functionvia revised Euler-Maclaurin-Siegel andAbel-Plana summation formulas
Xiao-Jun Yang , , , ∗ State Key Laboratory for Geomechanics and Deep Underground Engineering,China University of Mining and Technology, Xuzhou 221116, P. R. China College of Mathematics, China University of Mining and Technology, Xuzhou221116, P. R. China College of Mathematics and Systems Science, Shandong University of Scienceand Technology, Qingdao 266590, P. R. China
Abstract
This paper addresses the revised Euler-Maclaurin-Siegel and Abel-Plana summa-tion formulas and proves the Riemann hypothesis with the aid of the critical stripand the Todd type functions for the first time. The distribution formulae of theprime numbers and the twin prime numbers are discussed in detail. We also presentthe proof that all prime numbers are not less than 1. The results may be as accu-rately and efficiently mathematical approaches provided to open up the dawn of anew age in analytic number theory.
Key words:
Euler-Maclaurin-Siegel summation formula, Abel-Plana summationformula, Riemann’s Zeta function, Riemann hypothesis, prime, twin prime.
MSC 2010:
The Riemanns’s Zeta function (RZF) (also called as the Euler-Riemann Zetafunction) has been played an important role in the study on the prime numbers[1,2]. As a matter of fact, the important work of the RZF defined on the setof real numbers was firstly proposed in 1749 by Euler [3] and widely extended ∗ Corresponding author: Tel.: 086-67867615; E-Mail: [email protected] [email protected] (X. J. Yang)
Preprint submitted to Preprint: Arxiv 16 November 2018 y Chebyshev [4] and Riemann [5] in the complex domain [6,7], respectively.As one of the important tasks involving the RZF set up in the set of complexnumbers, the zeros of the RZF, which is defined as ([8,9,10,11,12,13,14]) ζ ( s ) = ∞ X n =1 n − s , (1)where s ∈ C , s = Re ( s )+ iIm ( s ) = σ + ib ∈ C , with i = √− Re ( s ) = σ ∈ R and Im ( s ) = b ∈ R , are considered as follows:(A1) The RZF has the trivial zeros at s = − n , where n ∈ N (see [8]);(A2) The RZF has the non-trivial zeros at the complex numbers s = 0 . ib ,where b ∈ R , and has the critical strip, e.g., { s ∈ C : 0 < Re ( s ) < } (see, i.e.,[15,16,17,18,19,20,21,22,23,24,25]);(A3) Re ( s ) = σ = 0 . In this section, we introduce the main results for the RZF and TF and theEuler-Maclaurin-Siegel [34,45,46,47] and Abel-Plana summation [48,49,50] for-mulas and their extensions.Let the symbols of the complex numbers, real numbers, integral numbers,natural numbers and rational numbers be C , R , Z , N and Q , respectively.2 .1 The theorems for the RZF Theorem 1 (see [8])The RZF is an analytic continuation to s ∈ C except s = 1 . Theorem 2 (see [8])The RZF has the trivial zeros at s = − n , n ∈ N . Theorem 3 (see [15,16,17,18,19,20,21,22,23,24,25])The RZF has the critical strip, e.g., { s ∈ C : 0 < σ < } . Theorem 4
If the RZF is an analytic continuation to all complex plane s ∈ C except s = 1 and s = − n with n ∈ N , then we have πi Z R ζ (1) ( s ) ζ ( s ) = ℘, (2) where ℘ is the number of the nontrivial zero points s interior to R = { s ∈ C , s = 1 , s = − n, n ∈ N } such that ζ (1) ( s ) = L = 0 . (3) Proof . With the use of the expression (see [53]), we have the result. This isgeneralized version referred to [9].
The TF is defined as (see [51]) T ( x ) = n X j =0 T j ( x j ) = d Y i =1 x i − e − x i , (4)where x ∈ C , x i ∈ C , x j ∈ C , T j ( x j ) ∈ C , i ∈ N , j ∈ N , n ∈ N and d ∈ N .If there is T ( x ) = Y i =1 x i − e − x i = 1 e πx − , (5)3here x ∈ C , then TF can be written as follows (see [52]): T ( x ) := T ( x ) = L ( x ) = Y i =1 x i − e − x i = 1 e πx − . (6)If there exists (see [52]) T j ( x j ) = L j ( x ) = d Y i =1 x i − e − x i = ( − j ∞ X k =1 e − πkx (2 πk ) j , (7)then TF can be given as follows: T ( x ) = ∞ X j =0 ( − j ∞ X k =1 e − πkx (2 πk ) j , (8)where i ∈ N , j ∈ N , k ∈ N , n ∈ N and d ∈ N . We also observe that L j ( x ), as Todd classes or TF, is analytic in C \ i Z andhas simple poles at x ∈ i Z [52]. For more details of the properties of the TF,see [51,52]. Lemma 1 (Euler-Maclaurin summation formula) (see [45,46,47,50,52])If f ( t ) has a continuous derivative f (1) ( t ) in the interval ≤ t ≤ , then wehave R f (1) ( t ) (cid:16) t − (cid:17) dt = f (0)+ f (1)2 − R f ( t ) dt. (9) Lemma 2 (Euler-Maclaurin-Siegel summation formula) (see [45,46,47,50,52];also see [34], p.2, formula 2)If f ( t ) is a complex-valued function and has a continuous derivative f (1) ( t ) in the interval ≤ t ≤ , then we have R f (1) ( t ) (cid:16) t − (cid:17) dt = f (0)+ f (1)2 − R f ( t ) dt. (10) Theorem 5 (Revised Euler-Maclaurin summation formula)Let f ( t ) have a continuous derivative f (1) ( t ) in the interval < t < . If f (0) = f ( t ) − t Z f (1) ( t ) dt → ∞ (11)4 nd f (1) = Z t f (1) ( t ) dt + f ( t ) → ∞ , (12) then there exists R f (1) ( t ) (cid:16) t − (cid:17) dt = R t f (1) ( t ) dt − t R f (1) ( t ) dt ! + f ( t ) − R f ( t ) dt. (13) Proof . With the use of Eqs.(11) and (12), from Eq.(1) we obtain the result.
Theorem 6 (Revised Euler-Maclaurin-Siegel summation formula)Let f ( t ) be a complex-valued function and have a continuous derivative f (1) ( t ) in the interval < t < . If Eqs.(11) and (12) are valid, then we have R f (1) ( t ) (cid:16) t − (cid:17) dt = R t f (1) ( t ) dt − t R f (1) ( t ) dt ! + f ( t ) − R f ( t ) dt. (14) Proof . Substituting Eqs.(11) and (12) into Eq.(2) we give the result.
Theorem 7 (see [52,54])If f ( z ) is continuous in the vertical trip ≤ u ≤ , holomorphic in its interior,and f ( z ) = O (cid:16) e π | χ ( z ) | (cid:17) as | χ ( z ) | → ∞ in the trip, uniformly with the respectto u , then the Abel-Plana formula states that R f (1) ( u ) (cid:16) u − (cid:17) du = f (0)+ f (1)2 − R f ( u ) du + M ( θ ) , (15) where z = u + iθ with u ∈ R and θ ∈ R . For 0 ≤ u ≤ θ ) as (see[48,49,50]) M ( θ ) = i ∞ Z f ( iθ ) − f ( − iθ ) e πθ − dθ, (16)the Abel-Plana representation of M ( θ ) as (see [52,54])M ( θ ) = i ∞ R f ( iθ ) − f ( − iθ )] − [ f (1+ iθ ) − f (1 − iθ )] e πθ − dθ, (17)and the alternative Euler-Maclaurin representation of M ( θ ) as (see [45,46,47])M ( θ ) = h f (1) (1) − f (1) (0) i − ∞ R (cid:18) ∞ P k =1 e − πkiθ (2 πk ) + ∞ P k =1 e πkiθ (2 πk ) (cid:19) f (2) ( θ ) dθ. (18)5e remark that (see [52,54]) f ( z ) = O (cid:16) e π | χ ( z ) | (cid:17) as | χ ( z ) | → ∞ implies that f ( n ) → n → ∞ , where n ∈ N . Theorem 8 (Revised Abel-Plana summation formula)Suppose that f ( z ) is continuous in the vertical trip < u < , holomorphic inits interior, and f ( z ) = O (cid:16) e π | χ ( z ) | (cid:17) as | χ ( z ) | → ∞ in the trip, uniformly withthe respect to u . If Eqs.(11) and (12) are valid, then the Abel-Plana formulastates that R f (1) ( u ) (cid:16) u − (cid:17) du = R u f (1) ( u ) du − u R f (1) ( u ) du ! + f ( u ) − R f ( u ) du − M ( θ ) , (19) where z = u + iθ with u ∈ R and θ ∈ R .Proof . Substituting Eqs.(11) and (12) into Eq.(5), we obtain the result. Theorem 9
Suppose that f ( z ) is continuous in the vertical trip < u < ,holomorphic in its interior, and f ( z ) = O (cid:16) e π | χ ( z ) | (cid:17) as | χ ( z ) | → ∞ in thetrip, uniformly with the respect to u . If Eqs.(11) and (12) are valid, then wehave M ( θ ) = 0 , (20) e.g., i ∞ R f ( iθ ) − f ( − iθ ) e πθ − dθ = 0 , (21) i ∞ R f ( iθ ) − f ( − iθ )] − [ f (1+ iθ ) − f (1 − iθ )] e πθ − dθ = 0 , (22) h f (1) (1) − f (1) (0) i − ∞ R (cid:18) ∞ P k =1 e − πkiθ (2 πk ) + ∞ P k =1 e πkiθ (2 πk ) (cid:19) f (2) ( θ ) dθ = 0 , (23) where z = u + iθ with u ∈ R and θ ∈ R .Proof . Compared among Eqs.(9), (10) and (13), we get the results. In this section we propose the proof of the RH making use of the above results.
Theorem 10 (RH) [8,23,24,26,31,34] et ζ ( s ) be an analytic continuation to s ∈ C and s = 1 . Let ζ ( s ) = 0 andlet s = σ + ib be a non-trivial zero, where σ ∈ R and b ∈ R . Then we have Re ( s ) = σ = .Proof . Family 1
We give the proof of the RH with the revised Euler-Maclaurin-Siegelsummation formula.Let ζ ( s ) = ζ ( σ + ib ) , where σ ∈ R and b ∈ R .Let b be a fixed number, e.g., ζ ( σ + ib ) = ζ ( σ ) , where ≤ σ ≤ .It is clear that ζ ( σ ) has a continuous derivative ζ (1) ( σ ) in the interval ≤ σ ≤ since, when b = 0 and σ = 0 we give ζ (0) = 0 and, when b = 0 and σ = 1 , ζ ( s ) has the simple pole at s = 1 .In view of Theorem 6 we now structure the function, defined as Ω ( σ ) = R σ ζ (1) ( σ ) dσ − σ R ζ (1) ( σ ) dσ ! + ζ ( σ ) − R ζ ( σ ) dσ − R ζ (1) ( σ ) (cid:16) σ − (cid:17) dσ, (24) which deduces from Eq. (14) that Ω ( σ ) = 0 . (25) Suppose that s = σ + ib is the nontrivial zero, then we always give ζ ( σ ) = 0 (26) and Z ζ ( σ ) dσ = 0 , (27) where ≤ σ ≤ and b ∈ R .Substituting Eqs.(26) and (27) into Eq.(25), we have R σ ζ (1) ( σ ) dσ − σ R ζ (1) ( σ ) dσ ! − R ζ (1) ( σ ) (cid:16) σ − (cid:17) dσ = 0 , (28) where ≤ σ ≤ and b ∈ R .With the help of Eq.(3), we have ζ (1) ( σ ) = L = 0 , (29)7 here ≤ σ ≤ and b ∈ R .Combining Eqs.(28) and (29), we arrive at R σ Ldσ − σ R Ldσ ! − R L (cid:16) σ − (cid:17) dσ = 0 , (30) which can be rewritten as follows: R σ Ldσ − σ R Ldσ ! = R L (cid:16) σ − (cid:17) dσ. (31) The left part of Eq.(31) can be expressed by R σ Lσdσ − σ R Ldσ ! = L R σ dσ − σ R dσ ! = L (1 − σ )= L (cid:16) − σ (cid:17) . (32) The right part of Eq.(31) can be represented in the form R L (cid:16) σ − (cid:17) dσ = L R σdσ − R dσ ! = L R σdσ − R dσ ! = 0 . (33) It follows from Eqs.(31), (32) and (33) that L (cid:18) − σ (cid:19) = 0 , (34) which deduces σ = 12 . (35) Family 2
We present an alternative proof of the RH involving the revisedAbel-Plana summation formula.It follows from Theorem 8 that R σ ζ (1) ( σ ) dσ − σ R ζ (1) ( σ ) dσ ! + ζ ( σ ) − R ζ ( σ ) dσ − R ζ (1) ( σ ) (cid:16) σ − (cid:17) dσ − M ( θ ) = 0 , (36) where ≤ σ ≤ and b ∈ R . ith the use of Eq.(20), Eq.(36) becomes R σ ζ (1) ( σ ) dσ − σ R ζ (1) ( σ ) dσ ! + ζ ( σ ) − R ζ ( σ ) dσ − R ζ (1) ( σ ) (cid:16) σ − (cid:17) dσ = 0 , (37) which is in agreement with Eq.(24).In a similar manner, we may obtain the same result.Thus we finish the proof of the RH. To give the description for the distribution of the prime numbers and the twinprime numbers, we begin with the following theorems.
Theorem 11 If s = + ib is the nontrivial zero of the RZF, e.g., ζ ( s ) = ζ (cid:16) + ib (cid:17) = 0 , where b ∈ R , then we have i ∞ R ζ ( ib ) − ζ ( − ib ) e πb − db = 0 , (38) i ∞ R ζ ( ib ) − ζ ( − ib )] − [ ζ (1+ ib ) − ζ (1 − ib )] e πb − db = 0 , (39) h ζ (1) (1) − ζ (1) (0) i − ∞ R (cid:18) ∞ P k =1 e − πkib (2 πk ) + ∞ P k =1 e πkib (2 πk ) (cid:19) ζ (2) ( b ) db = 0 , (40) and i ∞ R ζ (1+ ib ) − ζ (1 − ib )] e πb − db = 0 , (41) where ζ (0 + ib ) = ζ ( ib ) .Proof . Applying Theorem 9 and taking θ = b and f ( iθ ) = ζ ( ib ), we easilygive Eqs.(38), (39) and (40).Compared between Eqs.(38) and (39), we conclude Eq.(41).Thus, we finish the proof. Theorem 12 If ζ ( s ) has the nontrivial zero, then we have − s = 1 − σ − ib (42)9 uch that − s is the nontrivial zero of ζ ( s ) , where s = σ + ib with σ ∈ R and b ∈ R .Proof . With the use of Theorem 10 and s = σ + ib and b = 0, we have ∞ X n =1 e − s ln n = ∞ X n =1 e − ( σ + ib ) ln n = ∞ X n =1 e − σ ln n [cos ( b ln n ) − i sin ( b ln n )] , (43) ζ ( ib ) = ∞ X n =1 e − ib ln n = ∞ X n =1 cos ( b ln n ) − i ∞ X n =1 sin ( b ln n ) (44)and ζ ( − ib ) = ∞ X n =1 e ib ln n = ∞ X n =1 cos ( b ln n ) + i ∞ X n =1 sin ( b ln n ) . (45)Making use of Eqs.(44) and (45), we have by Eq.(38) that i ∞ Z ζ ( ib ) − ζ ( − ib ) e πb − db = 2 ∞ X n =1 ∞ Z sin ( b ln n ) e πb − db = 0 . (46)With the aid of Eq.(41), we get i ∞ Z ζ (1 + ib ) − ζ (1 − ib ) e πb − db = 2 ∞ X n =1 ∞ Z e − ln n sin ( b ln n ) e πb − db = 0 . (47)With the connection between Eqs.(46) and (47), we carry out ∞ X n =1 ∞ Z sin ( b ln n ) e πb − db = ∞ X n =1 ∞ Z e − ln n sin ( b ln n ) e πb − db, (48)which implies that ∞ X n =1 ∞ Z e − ln n − e πb − b ln n ) db = 0 (49)and sin ( b ln n ) = 0 , (50)where b ∈ R and n ∈ N .Considering s = + ib and s = − ib and using Eq.(50), we deduce that ζ (cid:18)
12 + ib (cid:19) = ∞ X n =1 e − ( + ib ) ln n = ∞ X n =1 e − ln n cos ( b ln n ) = 0 , (51)and ζ (cid:16) − ib (cid:17) = ∞ P n =1 e − ( − ib ) ln n = ∞ P n =1 e − ln n cos ( b ln n ) = 0 . (52)10oreover, we also give ζ (cid:18)
12 + ib (cid:19) = ∞ X n =1 e − ln n +2 πγi (53)and ζ (cid:18) − ib (cid:19) = ∞ X n =1 e − ln n − πγi (54)since sin ( b ln n ) = 0, which leads to cos ( b ln n ) = 1, where b ln n = 2 πγ (55)and cos ( b ln n ) = e πγi (56)with γ ∈ Z and n ∈ N .When γ ∈ Z / n ∈ N /
1, Eq.(55) is can be written as b = 2 πγ ln n . (57)According to the Conrey and Ghosh’s result [55], e.g., b = 2 πn ln q , (58)where q is a prime number, we have2 πγ ln n = 2 πn ln q (59)such that ln q = nγ ln n, (60)which leads to the distribution of the difference of any two primes ∆ qq = γ (cid:16) (ln n + 1) ∆ n − n ln nγ ∆ γ (cid:17) , (61)where ∆ q is the difference of any two primes, ∆ n ∈ Z is the difference oftwo numbers, and ∆ γ ∈ Z is the difference of two numbers. The table of thedistribution of the prime with respect to n and γ (all prime numbers less than100) is listed in Table 1.The distribution of the difference of any two prime numbers (for any primeand 3) is listed in Table 2.Let q and q be two prime numbers. From Eq.(61) we have ∆ q = q − q = 2[56] such that 2 q = ∆ nγ − nγ ∆ γ ! ln n + ∆ nγ , (62)11here ∆ n ∈ Z and ∆ γ ∈ Z .In a similar way, we always have ∆ q = q − q = − − q = ∆ nγ − nγ ∆ γ ! ln n + ∆ nγ , (63)where ∆ n ∈ Z and ∆ γ ∈ Z .It is seen that Eqs.(62) and (63) are used to describe the distributions ofthe twin prime numbers and the distributions of some twin prime numbers(all twin prime numbers less than 100) are shown in Table 3 and Table 4,respectively.Thus, from Eq.(60) one writes the distribution of the prime numbers withrespect to n and γ as q = e nγ ln n , (64)which leads to q = n nγ = ( n n ) γ and lim γ → + ∞ n ∈ N q = 1 . (65)With the aid of Eqs.(57) and (58), the distribution of the prime numbers withrespect to b and γ can be given as follows: q = e πb e πγb and lim b → + ∞ γ ∈ N q = 1 . (66)Eqs.(65) and (66) implies that all prime numbers are not less than 1.Therefore, we obtain the difference of any two primes, e.g.,∆ q = 4 π b e πγb [2 π ∆ γ − (1 + γ ) ∆ b ] , (67)where ∆ q is the difference of any two primes, ∆ γ ∈ Z is the difference of twonumbers and ∆ b ∈ R is the difference of the variable b .From Eq.(66) we easily observe that the number of prime numbers is infiniteand belong to a finite set.For ∆ q = q − q = 2 [56] and ∆ q = q − q = − π b e πγb [2 π ∆ γ − (1 + γ ) ∆ b ] (68)and − π b e πγb [2 π ∆ γ − (1 + γ ) ∆ b ] , (69)12hich can be represented as the distributions of the twin prime numbers.When γ takes the values for the natural numbers from 1 to 20, the distributionof the prime numbers with respect to b and γ is illustrated in Figure 1.With the aid of Eqs.(57) and (58), one may get γ = n ln n ln q , (70)which leads to the distribution of the difference of any two primes, e.g.,∆ γ = n ln nq ∆ q + ln n ln q ∆ n + ∆ n ln q , (71)where ∆ q is the difference of any two primes, ∆ n ∈ Z is the difference of twonumbers, and ∆ γ ∈ Z is the difference of two numbers.For ∆ q = q − q = 2 [56] we have∆ γ = 2 n ln nq + ln n ln q ∆ n + ∆ n ln q , (72)where ∆ n ∈ Z is the difference of two numbers, ∆ γ ∈ Z is the difference oftwo numbers and ∆ b ∈ R is the difference of the variable b .Similarly, for ∆ q = q − q = − γ = − n ln nq + ln n ln q ∆ n + ∆ n ln q , (73)where ∆ n ∈ Z is the difference of two numbers, ∆ γ ∈ Z is the difference oftwo numbers and ∆ b ∈ R is the difference of the variable b .Note that Eqs.(60), (64), (65), (66) and (70) are equivalent as the alternativedescriptions of the distribution of the prime numbers and that Eqs.(62), (63),(67), (68), (69), (72) and (73) may be presented to find the distribution of thetwin prime numbers.From Eq.(62) we have ∆ q = q − q = 2 [56] and q = 6 χ − χ ∈ Nsuch that q = 6 χ + 1 (74)and 26 χ − γ ∆ n − nγ ∆ γ ! ln n + ∆ nγ , (75)where ∆ n ∈ Z and ∆ γ ∈ Z . 13rom Eq.(75) we may give χ = 26 γ (cid:16) ∆ n − n ∆ γγ (cid:17) ln n + ∆ n + 16 , (76)where n ∈ N , γ ∈ N , ∆ n ∈ Z and ∆ γ ∈ Z .Taking γ ∆ n = n ∆ γ , Eq.(76) can be represented in the form: χ = 26 δ + 16 , (77)which implies that δ = γ ∆ n (78)and λ = 2 δ, (79)where λ ∈ N , γ ∈ N , ∆ n ∈ Z and δ ∈ Q .The distribution of the prime number χ with respect to γ and ∆ n is listed inTable 5.It is clear that all twin prime pairs of the forms (6 χ − , χ + 1) and ( λ, λ + 2)are true since Eq.(77) holds; it is to say, χ ∈ N and the twin prime pairs(6 χ − , χ + 1) and ( λ, λ + 2) are infinite and belong to finite set [56]. From Eqs.(57), (58) and (59) we have s = 12 + i πγ ln n (80)and s = 12 + i πn ln q (81)such that ζ (cid:18)
12 + ib (cid:19) = 0 (82)with b = 2 πγ ln n = 2 πn ln q . (83)Similarly, there exist s = 12 − i πγ ln n (84)14nd s = 12 − i πn ln q (85)such that ζ (cid:18) − ib (cid:19) = 0 (86)with Eq.(81).The relationship among b , γ , n and q (all prime numbers less than 100) andthe distribution of the non-trivial zeros of the RZF with respect to q (all primenumbers less than 100) are listed in Table 6 and Table 7, respectively. For themore details for the computations of the RZF, see [58]. In the present work, we structured the functions within the revised Euler-Maclaurin-Siegel and Abel-Plana summation formulas through the criticalstrip and Todd function. Next, we presented the RH, which states that thereal part of every non-trivial zero of the RZF is 0.5. The distribution repre-sentations of the prime numbers and the twin prime numbers were obtainedin detail. Moreover, the distributions of the differences of the two prime num-bers and two twin prime numbers were also discussed in detail. Finally, weobserved that the prime numbers and twin prime numbers are infinite andbelong to finite set and the distributional laws of the twin prime numbers,e.g., (6 χ − , χ + 1) and ( λ, λ + 2) hold. We gave the explanation that whyall prime numbers are not less than 1. The obtained results are important toprovide to open up the dawn of a new age in analytic number theory. References [1] Hardy, G. H., Littlewood, J. E. (1916). Contributions to the theory of theRiemann zeta-function and the theory of the distribution of primes. ActaMathematica, 41(1), 119-196.[2] Jessen, B., Wintner, A. (1935). Distribution functions and the Riemann zetafunction. Transactions of the American Mathematical Society, 38(1), 48-88.[3] Euler, L. (1749). Sur la perfection des verres objectfs des lunettes. M´emoiresde l’acad´emie des sciences de Berlin, 274-296.[4] Chebyshev, P. L. (1946). Selected mathematical works, Moscow-Leningrad (InRussian).
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56] Hardy, G. H., Wright, E. M. (1979). An introduction to the theory of numbers.Oxford university press.[57] Caldwell, Chris K. Are all primes (past 2 and 3) of the forms 6n+1 and 6n-1?. The Prime Pages. The University of Tennessee at Martin. Retrieved 2018-09-27.[58] Bober, J. W., Hiary, G. A. (2018). New computations of the Riemann Zetafunction on the critical line. Experimental Mathematics, 27(2), 125-137. ig. 1. The distribution of the prime numbers with respect to b and γ . able 1The distribution of the primes. ln q = nγ ln nq n γ able 2The distribution of the difference of any two prime numbers q and q . ∆ qq = γ (cid:16) (ln n + 1) ∆ n − n ln nγ ∆ γ (cid:17) q q = q n γ ∆ q = q − q ∆ n ∆ γ able 3The distribution of the twin prime numbers. q = (cid:16) ∆ nγ − nγ ∆ γ (cid:17) ln n + ∆ nγ q n γ ∆ n ∆ γ − q = (cid:16) ∆ nγ − nγ ∆ γ (cid:17) ln n + ∆ nγ q n γ ∆ n ∆ γ able 5The distribution of the twin prime numbers. χ = δ + The twin prime pairs The twin prime pairs(6 χ − , χ + 1) ( λ, λ + 2) χ χ χ − χ + 1 2 δ δ + 21 1 6 5 7 5 72 2 12 11 13 11 133 3 18 17 19 17 194 5 30 29 31 29 315 7 42 41 43 41 436 10 60 59 61 59 617 12 72 71 73 71 73 able 6The relationship among b , γ , n and q (all prime numbers less than 100). q n γ b πln πln πln πln πln πln πln πln πln
10 29 29 29 πln
11 31 31 31 πln
12 37 37 37 πln
13 41 41 41 πln
14 43 43 43 πln
15 47 47 47 πln
16 53 53 53 πln
17 59 59 59 πln
18 61 61 61 πln
19 67 67 67 πln
20 71 71 71 πln
21 73 73 73 πln
22 79 79 79 πln
23 83 83 83 πln
24 89 89 89 πln
25 97 97 97 πln able 7The distribution of the non-trivial zeros of the RZF with respect to n , γ and q (allprime numbers less than 100) b = 2 πγ/ ln n = 2 πn/ ln q s − sq b . ib . − ib π/ ln 2 0 . i π/ ln 2 0 . − i π/ ln 22 3 6 π/ ln 3 0 . i π/ ln 3 0 . − i π/ ln 33 5 10 π/ ln 5 0 . i π/ ln 5 0 . − i π/ ln 54 7 14 π/ ln 7 0 . i π/ ln 7 0 . − i π/ ln 75 11 22 π/ ln 11 0 . i π/ ln 11 0 . − i π/ ln 116 13 26 π/ ln 13 0 . i π/ ln 13 0 . − i π/ ln 137 17 34 π/ ln 17 0 . i π/ ln 17 0 . − i π/ ln 178 19 38 π/ ln 19 0 . i π/ ln 19 0 . − i π/ ln 199 23 46 π/ ln 23 0 . i π/ ln 23 0 . − i π/ ln 2310 29 58 π/ ln 29 0 . i π/ ln 29 0 . − i π/ ln 2911 31 62 π/ ln 31 0 . i π/ ln 31 0 . − i π/ ln 3112 37 74 π/ ln 37 0 . i π/ ln 37 0 . − i π/ ln 3713 41 82 π/ ln 41 0 . i π/ ln 41 0 . − i π/ ln 4114 43 86 π/ ln 43 0 . i π/ ln 43 0 . − i π/ ln 4315 47 94 π/ ln 47 0 . i π/ ln 47 0 . − i π/ ln 4716 53 106 π/ ln 53 0 . i π/ ln 53 0 . − i π/ ln 5317 59 118 π/ ln 59 0 . i π/ ln 59 0 . − i π/ ln 5918 61 122 π/ ln 61 0 . i π/ ln 61 0 . − i π/ ln 6119 67 134 π/ ln 67 0 . i π/ ln 67 0 . − i π/ ln 6720 71 142 π/ ln 71 0 . i π/ ln 71 0 . − i π/ ln 7121 73 146 π/ ln 73 0 . i π/ ln 73 0 . − i π/ ln 7322 79 158 π/ ln 79 0 . i π/ ln 79 0 . − i π/ ln 7923 83 166 π/ ln 83 0 . i π/ ln 83 0 . − i π/ ln 8324 89 178 π/ ln 89 0 . i π/ ln 89 0 . − i π/ ln 8925 97 194 π/ ln 97 0 . i π/ ln 97 0 . − i π/ ln 97ln 97