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The Problems Existing in the Complex Continuations of the Gama Function and the Euler Formula of Prime Numbers

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LJRS Volume 21, Issue 2, Compilation 1.0

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Attribution 2.0 Generic (CC BY 2.0)

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Abstract

Abstract The real Gama Function ‹?(a) is defined in the field of real number with a ‹?? 0 . It is infinite and meaningless in the field with a ‹?¬ 0 . However, according to the continuation theory of function at present, except at the points a ‹?« 0, ‹??1, ‹??2, ‹??3,‹Ÿ?‹Ÿ?‹Ÿ? , ‹?(a) can be extended to the field of negative number with a ‹?¬ 0 . It is pointed that this continuation leads to infinite and is meaningless. The reason is that the form of Gama function has no any change in the extended field, so that contradiction is caused. It is proved that the single complex continuation of Gama function does not satisfy the Cauchy-Riemann equation, so it is not an analytic function. But the double complex continuation of Gama function satisfies the Cauchy-Riemann equation and isan analytic function. In addition, it is proved that the complex continuation of complementary formula of Gama function is incorrect. The correct result is calculated in the paper. The influences of these results on the Riemann Zeta function and the Riemann hypotheses are discussed. According to correct formula, the zeros of Zeta function equation are located on the points a ‹?« ‹?ñ1/ 2, ‹?ñ3/ 2, ‹?ñ5 / 2‹Ÿ?‹Ÿ?‹Ÿ? , so the Riemann hypothesis does not hold. The complex continuation of Euler formula of prime numbers is also discussed. It is proved that the real part of the complex continuation formula is different from the original Euler formula of prime number. Because, the trigonometric functions are contained in it which my be irrational numbers, the extended formula does not describer the relationnbetween nature numbers and prime numbers again. Therefore, the complex continuation of Euler formula of prime numbers is meaningless in number theory. We can not discuss the contribution of prime numbers based on it.

The Problems Existing in the Complex Continuations of Gama Function and the Euler Formula of Prime Numbers

Mei Xiaochun

ABSTRACT

The real Gama Function  is defined in the field of real number with . It is infinite and meaningless in the field with . However, according to the continuation theory of function at present, except at the points , can be extended to the field of negative number with . It is pointed that this continuation leads to infinite and is meaningless. The reason is that the form of Gama function has no any change in the extended field, so that contradiction is caused. It is proved that the single complex continuation of Gama function does not satisfy the Cauchy-Riemann equation, so it is not an analytic function. But the double complex continuation of Gama function satisfies the Cauchy-Riemann equation and is an analytic function. Also, it is proved that the complex continuation of complementary formula of Gama function is incorrect. The correct result is calculated in the paper. The influences of these results on the Riemann Zeta function and the Riemann hypotheses are discussed. According to the correct formula, the zeros of Zeta function equation are located on the points , so the Riemann hypothesis does not hold. The complex continuation of the Euler formula of prime numbers is also discussed. It is proved that the real part of the complex continuation formula is different from the original Euler formula of prime numbers. Because the trigonometric functions are contained in it, which my be irrational numbers, the extended formula does not describe the relation between natural numbers and prime numbers again. Therefore, the complex continuation of the Euler formula of prime numbers is meaningless in number theory. We can not discuss the distribution of prime numbers based on it.

Keywords: gama function, negative continuation of function, complex continuation of gama function, analysis function, residue theorem, euler formula of prime number, riemann hypothesis, riemann zeta function equation, cauchy-riemann equation.

Author: Department of Theoretical Physics and Pure Mathematics, Institute of Innovative Physics in Fuzhou, Fuzhou, China.

INTRODUCTION

It was revealed in the author’s first paper [1] that there were several serious mistakes in the Riemann’s original paper to deduce the Zeta function equation in 1859.

  1. An integral item around the original point of the coordinate system was neglected when Riemann deduced the integral form of the Riemann Zeta function. This item is convergent when but divergent when . The integral form of the Zeta function has not changed the divergence of its series form.
  2. A summation formula was used in the deduction of the integral form of the Zeta function. The applicable condition of this formula is . At point , the formula is meaningless. However, the lower limit of the Zeta function’s integral is , so the formula can not be used.
  3. The formula  of Jacobi function was used to prove the symmetry of the Zeta function equation. The applicable condition of this formula is also . Because the lower limit of the integral is , this formula can not be used too.
  4. Due to these mistakes, the inconsistency is caused on the two sides of the Zeta function equation. On the real axis of complex plane, the function equation holds only at point . However, at this point, the Zeta function is infinite, rather than zero. At other points of the real axis with  and , the two sides of the Zeta function equation are contradictory. When one side is finite, another side may be infinite.

Therefore, the Riemann Zeta function equation does not hold, and the Riemann hypothesis is meaningless. This is the fundamental reason why the Riemann hypothesis has not been proved for so many years. At present, the zero calculations of the Zeta function are approximate due to the series with different orders are used. The analytic property of the Zeta function is destroyed so that the so-called zeros are not real ones.

In the author’s second paper [2], the Zeta function equation is assumed to be correct, a simple and strict method is proposed to prove that the Riemann Zeta function equation has no trivial zero. The method is to divide the real part and the imaginary part of the Zeta function equation completely. By comparing the real part and the imaginary part individually, an equation set about and is obtained.

It is proved that the only solution of this equation set is  and . So  is not the non-trivial zero of the Zeta function equation. If , four equations are obtained. It is proved that these four equations are not independent of each other. To make them independent, the only way is to let  and . However, in this case, we have , rather than zero. Finally, the comparing method of infinite series is used to prove that the Zeta function itself has no zero. Therefore, the Riemann hypothesis does not hold.

Because the Riemann Zeta function equation is based on the Gama function, the negative and complex continuations of the Gama function are discussed in this paper. To do it, we need to define the continuation of a function clearly at first. Assume that the function  has a clear definition in the field  but it is meaningless in the field . To make it meaningful in the field , the function should be extended into . The analytic continuation of a function should satisfy the following conditions. 

  1. After the continuation, the form of the function  in the field  should have something different from the function . Otherwise, the contradiction would be caused, and the continuation became meaningless.  
  2. In the original field , the form of function  must be completely the same as the form of function , otherwise  can not be regarded as the continuation of .
  3. Just for the sake of uniqueness, the continuation of function  must be analytic, or the derivative of function  exists everywhere. If it is a complex continuation, the Cauchy-Riemann equation must be satisfied.

The Gama function is a very important function with wide applications in pure mathematics and practical problems. Some people think that its importance is only next to the trigonometric function and the exponential function.

For example, based on the Gama function, the integral form of the Riemann Zeta function is introduced. It is the basis of Riemann hypothesis problem. It is proved in this paper that the negative continuation of the Gama function does not satisfy the condition 1 and leads to infinite. The single complex continuation of the Gama function does not satisfy the condition 3, so it is not an analytic function. But the double complex continuation of the Gama function satisfies all three conditions, so it is an analytic continuation.

It is proved that the present complex continuation of the Gama function complementary formula is wrong. It is the result directly let  in  . The correct calculation result is  

                                                                                                              (1)        

The result is a real number, rather than a complex number. The influences of these results on the Riemann Zeta function equation and Riemann hypotheses are discussed in this paper. It is proved that after Eq.(1) is used, the zeros of the Zeta function equation are located on the points , the Riemann hypothesis is proved invalid from another angle.  

The complex continuation of the Euler formula of prime numbers is discussed in this paper. It is proved that after the Euler formula of prime numbers is extended into the complex number field, the real part of the formula is different from the original Euler formula of prime numbers. Because it contains the trigonometric functions  and  which may be irrational numbers, the extended formula does not describe the relation between natural numbers and prime numbers again. Therefore, the complex continuation of the Euler prime number formula is meaningless in number theory. We can not discuss the distribution of prime numbers based on it.  

  1. THE CONTINUATIONS OF GENERAL FUNCTIONS

2.1 The analytic continuations of real functions

As we knew that a function has its definition field in general. Out of the field, the function may be meaningless. In order to make the function meaningful in the greater field, the continuation is needed. The common example of a real function’s continuation is

                                                                                      (2)

Where  is a real number. Eq. (1) is meaningful and limited in the field . when , the function  and becomes meaningless. On the other hand, we define

                                                                                                                  (3)

 is meaningful on the whole number axis except at the point . At all points of the field with , we always have . By developing  into the Taiwan's series when , we can prove  completely.

However, when , Eqs. (2) and (3) can not be equal each other. For example, let , we have  and . Because the definition field of  is greater than that of , we consider  as the continuation of  in the field with  and write two functions in the unified form as below

                                                                  (4)

In general, a function can be extended in many different ways, resulting in different results. In order to ensure the uniqueness, the continuation of function needs to meet the continuity condition, so that the function can be differentiated everywhere. The continuation that meets this condition is called as the analytic continuation [3].

It is important to emphasize that at every point in the small field where the original function is meaningful, the value of the extended function should be exactly the same as the value of original function, otherwise it is not the continuation of the original function. Meanwhile, in the extended field, the form of the extended function must be different from the original function, otherwise the extension of the function is meaningless [3].

For example, for Eqs. (2) and (3), in the field where the original function makes sense, the values of function and  must be the same at every point. Although their forms look different on the surface, they are the same actually. In the field  after the continuation, the forms of  and  must be different. Otherwise  is still equal to, which is no meaning.

It is difficult to find the form of analytic continuation of a function in practical problems. The analytic continuations of some functions in existing mathematics are actually unsuccessful. As we see below, the negative continuation of the Gama function violates the above principles. In the extended field, the form of the function is exactly the same as the original function’s form, so it is still infinite and meaningless.

2.2 The analytic continuation of complex function

Let  be a complex number, a similar example of analytic continuation of complex function is shown below.

         

                                                                                                         (5)

 is an analytic function and convergent inside the unit circle but is divergent outside the unit circle without meaning.  is an analytic function meaningful on the whole complex plane except at points ,

To developing  into the Taylor's series of complex functions in the field , we can obtain . That is to say, two functions are completely the same. Since the definition field of  is larger than , can be regarded as an analytic continuation of  over the entire complex plane (except at points ). Similar to Eq.(4), we write they in the unified form

                                                                      (6)

Similarly, within the extended field (), and are not the same functions, because  is meaningless in this case.

  1. THE NEGATIVE CONTINUATION OF THE REAL GAMA FUNCTION

3.1 The definition and character of the real Gama function

The original definition of the real Gama function is

                                                                              (7)

Here parameter  is a real number with . If , the integral of Eq.(7) is infinite and meaningless. The Gama function has the following natures [3]

                                                                      (8)

      or                                         (9)

      ()                                         (10)

                                                                   (11)

                                              (12)

According to the formulas above, the Gama function has no zero with in the field .

3. 2 The negative continuation of the Gama function does not hold.

According to the original definition, the Gama function is meaningless when . For example, when , Eq. (7) becomes

                                                      (13)

By the same method, it can be proved that

                                                                           (14)

                                                                          (15)

Therefore, when , we have .

However, according to the present theory of the Gama function, as long as , by using formula (9), Eq.(7) can still be extended from the field  to the field , called as the negative continuation of the Gama function with 

                                                       (16)

For example, let , according to Eqs. (7), (8) and (16), we have [3]

                                                 (17)

Let , according to Eqs.(16) and (17), we have

                                             (18)

So, according to the present theory, after negative continuation, as long as  is a non-negative integer, the Gama function is still meaningful. The formula (16) appears in various textbooks and literature, and be used widely in mathematics.

However, according to this direct negative continuation, the basic form of the Gama function has no any change. It violates the first condition and is wrong. For example, to take , according to the original definition Eq.(7) and Eq.(16), we have 

                                                    (19)

So we get the absurd result , or . So Eq.(16) can not hold. Again, to take , according to Eq.(16), we have

                                                     (20)

We still get , which is completely different from Eq.(18). In fact in the general situation, let , we have . According to Eqs.(7) and (16), the result is

                                       

         

                                                     (21)

That is . Therefore, the negative continuation formula (16) of the Gama function does not hold. The reason is that in the extend field, the form of the Gama function has not been changed.

  1. THE COMPLEX CONTINUATION OF THE GAMA FUNCTION.

4.1 The single complex continuation and double complex continuation of Gama function.

Let in Eq. (7), the Gama function is extended from real field to complex field with  

                                                                      (22)

We call Eq. (22) the single complex continuation of the Gama function. In the deduction of the integral form of the Riemann Zeta function, the single complex continuation of the Gama function is is involved.

According to the definition, the condition that Eq.(22) holds is . That is to say,   Eq.(22) holds only on the right half side of complex plane. On the left half side of complex plane, it does not hold. However, according to the present theory, Eq. (22) is actually considered to be valid on the whole complex plane except at the points of negative integers with . The reason is due to the following relation  

                                                                                      (23)

By considering the negative continuation with  in Eq.(24), the result is

                                             (24)

In this way, is extended into the negative half plane with . Similar to the negative continuation of the real Game function as shown in Eq.(16), we call Eq. (24) the negative continuation of complex Game function.  

By substituting  in Eq.(22) and separating the real part and the imaginary part, we have

     

           

                                                                                                          (25)

Here  and  are the real functions with

                            (26)

It can be seen that  and are not the originally defined Gama functions due to the existence of the items  and . A new parameter is increased in Eq. (26), so it becomes the function of double parameters. We can not call them the Gama function again. The formulas (8) ~ (12) can not hold for them. They need to be proved again. Only when , Eq.(25) degenerates to the real Gama function.

If let  again in Eq.(22), we call it the double complex continuation of the Gama function with

                                                                               (27)

In the deduction of the Riemann Zeta function equation, the double complex continuation of the Gama function is involved. We will discuss them in Section 4.4.  

4. 2 The formula of negative continuation of Gama function does not hold.

We now prove that the formula (24) of the negative continuation of the Gama function does not hold on the complex half-plane . Similarly, we have

 

                        

                                                         (28)

We have

                                    (29)

Let , so . When , we have ,,, but and are limited values. According to Eq.(28), we get the same result , or . So Eq. (24) does not hold when .

4. 3 The single complex continuation of the Gama function is not an analytic function

In the theory of complex function, the analytic property of function is very important. There are many theorems which are effective only for analytic functions, such as the Cauchy theorem, the residue theorem, and so on. These theorems do not work if the functions are not analytic ones.

To express the analytic nature, the real part and the imaginary part of complex function  should be separated. We write it as

                                                                       (32)

The real part and the imaginary part are not independent, both should satisfy the following Cauchy-Riemann equation [4]

                                                                               (33)

At present, the single complex continuation  of the real Gama function is considered an analytic function. But it is not true. We prove it below.

Let  and  in Eq.(25). If is an analytic function, following relation should be satisfied.

                                                                                (34)

However, according to Eq.(26), we have

                     (35)

                     (36)

It is obvious that Eq.(34) can not hold, so the Gama function  described by Eq.(25) is not an analytic function on the complex plane .

4. 4 The double complex continuation of the Gama function is an analytic function.

We prove below that the double complex continuation of the Gama function is an analytic function. We write Eq.(27) as

                                                      (37)

We have

                            (38)

                      (39)

                 (40)

                                               (41)

                                                  (42)

Here  and . Then, we write Eq.(37) as

                   (43)

and get

                                                                              (44)

                                              (45)

                                             (46)

From Eq.(41) and (42), we have

                                                 (47

                                                                                                   (48)

                                                (49)

                                              (50)

By considering Eqs.(47) ~ (50), we see that Eq.(33) can be satisfied. So Eq.(44) satisfies the Cauchy-Riemann equation, the double complex continuation of the Gama function is an analytic function.

4.5 The product of the Riemann Zeta function and the Gama function is not an analytic function.

The original definition of the series form of the Riemann Zeta function is

                                                                                                                          (51)

It is proved in [2] that Eq.(51) is an analytic function. Riemann obtained the relation between the Riemann Zeta function and the complex continuation of the Gama function in his original paper in 1859.

                                                                                                  (52)

We prove below that  is not an analytic function about . By using , Eq.(52) can be written as  

                                 (53)

By separating the real part and the imaginary part, we get

                              (54)

So we have

                                  (55)

                          (56)

It is obvious that Eqs.(55) and (56) are similar to Eqs.(35) and (36). The Cauchy-Riemann equation (33)can not be satisfied, so Eq.( analytic function 52) is not an analytic function.

4. 6 The double complex continuation of  function is an.

 Similar to Eq.(27), let  in Eq.(52) for the second complex continuation, we get

                                                                               (57)

It is proved that the double complex continuation  is an analytic function. We have

                                    (58)

                                            (59)

                   (60)

               (61)

                                                (62)

Similar to the calculations of Eqs.(47) ~ (50), it can be proved that Eq.(33) can be satisfied, so the function described by Eq.(57) is an analytic function. Riemann used it to calculate and deduce the Zeta function equation.  

  1. THE COMPLEX CONTINUATION OF COMPLEMENTARY FORMULA OF GAMA FUNCTION

5.1 The complex continuation of complementary formula of the Gama function does not hold

Suppose that  is a real non-integer number with , it can be proved to exist the following formula [4]

                                                                           (69)

Though the formula is defined in the real field, it needs to use the Residue theorem of complex function to prove. Let , the complex continuation of Eq.(69) is considered to directly let  in Eq.(69) at present and get

                                                                           (70)

Correspondingly, the complex continuation of Eq.(11) is considered to be

                                                                  (71)

Eq. (71) was used when Riemann deduced the integral form of the Zeta function in his original paper in 1859.  

It is proved below that Eq. (70) does not hold, so Eq.(71) does not hold too. We write the left side of Eq.(70) as

                                      (72)

According to the Euler formula, we have

                                                                   (73)

                                                             (74)

      (75)

If Eq.(70) holds, by comparing the real parts and the imaginary parts of Eq.(72) and (75), we have

                                   (76)

                                    (77)

Let  in Eqs.(76) and (77), we get

                                                      (78)

                                                             (79)

Eq.(78) is completely the same as Eq.(69), but Eq. (79) can not hold when . Where is wrong? Let’s analyze it below.

5.2 The correct calculation of complex continuation of complementary formula of the Gama function

At present, the Residue theorem is used to calculate Eq.(69). Let’s repeat this calculation at first. Suppose that  is a real non-integer, we consider the integral of complex function below with [4]

                                                                               (80)

Here  is a single value and analytic function everywhere except at several isolated singularities.  There are no singularities on the positive real axis. When  and ,  tends to zero consistently.

留数围道-1

                       Fig.1 The contour of residue calculation

The integral contour  of residue calculation is shown in Fig.1. It stars off from the point  on the above positive real axis, goes along the positive real axis and arrive at point A. then goes along a big circle with radius and comes back to point B on the down positive real axis. Then goes along the negative direction of real axis and arrives at point . At last, goes around the small circle  and reaches the starting point A.

Therefore, the integral of Eq.(80) can be written as

              (81)

By using the Residue theorem, we have

                       (82)

Because of  when  and , we have,

       and                    (83)

Based on the formula above, we get from Eq. (82)

            (84)

Let ,there is an unique singularity  on the real axis. The residue is

           (85)

Substituting it in Eq.(84), we get the current result with

           (86)

If let , it is impossible to let  on the right side of Eq.(86). It needs to be calculated again. Let

               (87)

                   (88)

When and , and  are uncertain values. But we still have  and , so the conditions  and  still hold. We can still use the Residue theorem. For , similar to Eq.(85), we have

            (89)

For , the result is

           (90)

The last result is

          (91)

          (92)

When , Eq.(91) is consistent with Eq.(78). Eq.(92) is equal to zero. The contradiction shown in Eq.(79) does not exist again. Eq.(70) becomes

       (93)

The result of integral is a real number, rather than a complex number. The complex continuation formula (71) should be changed as

                    (94)

It is also a real number, rather than a complex number.

It should be noted that Eq. (94) holds only when . Beyond this condition, Eq. (94) is still invalid. Eq. (71) was used when Riemann derived the Zeta function equation. These results will have a great influence on the Riemann hypothesis problem.

6.The complex continuation of gamma function multiplier product formula.

    The multiplier product formula of the Gama function is [4]

                   (95)

According to the current theory, the complex continuation of Eq.(95) is directly written as

                         (96)

Eq.(96) is regarded to be tenable on the whole complex plane. But there is no basis for this result actually. By using the definition of the Gama function, Eq.(95) can be written as

                 (97)

By substituting  in Eq.(96) and considering , the results are

 

 

 (98)

By separating the real part and the imaginary part, Eq.(98) becomes

 

   (99)

 

   (100)

The status of Eqs.(99) and (97) are equivalent, but they are different when . Eq.(97) needs proof but we have no at present. And so do for Eq.(100). So the complex continuation formula of the gamma function multiplier product is generally incredible. Because the double integrals are involved in the left sides of Eq.(99) and (100), the calculations are complex, they will be discussed in a separated paper.

7.The complex continuation of the Euler formula of prime numbers

   The Euler formula of prime numbers is

                         (101)

Here  is a real number, is nature number and is prime number. They are all positive integers. By considering , we have

                    (102)

By developing the multiple products on the right side of Eq.(101) , it can be written as

 

 

               (103)

Riemann extended the Euler formula of primes number into the complex field and write it as

                        (104)

Here  is a complex number. It should note that the Riemann’s complex continuation is only a hypothesis without strict proof. It is proved below that this complex continuation has no meaning in the theory of prime numbers for it does not describe the relation between prime numbers and natural numbers again.

     Let in Eq.(104), we have

                       (105)

By using , the left side of Eq.(105) can be written as

             (106)

The right side of Eq.(105) becomes

 

 

   

                     (107)

By separating the real part and the imaginary part of Eq.(105), we get

 

 

 

   (108)

 

  

 

 

   (109)

All quantities in Eqs.(108) and (109) are real numbers. Because  and  are not integers in general, we can regard them as new numbers. They may even be irrational numbers which can not be represented by the addition, subtraction, multiplication and division of two integers. So these two formulas do not describe the relation between nature numbers and prime numbers. Since both prime and natural numbers are integers, it is inconceivable that the connection between integers is established through irrational numbers.

The status of Eq.(108) and (103) are parallel, but they are obviously different. Eq.(103) was proved strictly by Euler. Eq.(108) was based on the complex continuation of Eq.(101) proposed by Riemann without any proof. Because Eq.(108) does not describe the relation between natural numbers and prime numbers, and contradicts with Eq.(103), it is meaningless. Eq.(109) has no basis and proof too. We has no reason to think it is correct.

Therefore, the complex continuation of the Euler formula of prime numbers has no meaning actually. The theory of prime numbers based on this formula can not be correct.

  1. The influence on the problem of the Riemann Hypothesis

8.1 The deductions of integral form of the Riemann Zeta function and the Zeta function equation

Based on the complex continuation form of the Gama function, Riemann deduced the integral form of Zeta function [1, 6]. Let  in Eq.(22), the result is

 

                             (110)

Taking the summation of Eq.(110), Riemann get

                                    (111)

Then, Riemann used the following summation formula of series (The original paper of Riemann had not provided this formula but used it actually.) [8].

                 

                             (112)

By substituting Eq.(112) in Eq.(11), Riemann obtained

                                                (113)

To calculate Eq.(113), Riemann extended the integral of real number  to the field of complex number with and defined the function

                                                       (114)

Eq. (114) is just the double complex continuation of Gama function. The definition domain of function  was extended into whole complex plane except at the point , rather than original .

黎曼猜想

                   Fig. 2 The integral path of Riemann Zeta function

The path of integral started from  to  along the straight line  under the axis. Here  was a small quantity. Then the path was along the circle  with radius  around the original point of the coordinate system. At last, the path was from  to  along the straight line  above the axis.

According to Fig. 2, Eq.(114) contains three items.

                                       (115)

Riemann’s paper provided the following result without concrete calculation [1]

                                                       (116)

It indicated that Riemann assumed that the medium item on the right side of Eq.(115) was zero

                                               (117)

By using the Euler’s formula

                                                        (118)

and substituting Eqs.(16) and (18) in Eq.(13), the result was

                                                        (119)

By using the complementary formula of the Gama function

                          (120)

and considering Eq. (114), Eq. (19) can be written as

                                     (121)

Riemann used the method of contour integral and the residue theorem, to obtain the integral  

                                        (122)

Eq.(121) becomes

                                    (123)

By considering the original definition of the Riemann Zeta function

                                                (124)

the Riemann Zeta function equation was obtained.

                                        (125)

Eq.(125) describes the relation between and. Then, Riemann introduced the transformation [1,4]

                                                (126)

and proved the follow symmetry relation

                               (127)

Since the functions  and have the same zero, based on Eq.(126), Riemann proposed the hypothesis that all non-trivial zeros of Zeta function lie on the straight line of .

At present, it is generally believed that after the Riemann complex continuation, the right side of Equation (125) becomes a new definition of Zeta function, so the left side of Eq.(125) is no longer in the form of Eq. (124). However, this is not the case because to get Eq. (125) from Eq. (123), Eq. (124) must be used. It must be remembered that the prototype of Eq. (125) is Eq. (123). Equation (125) is only a symbolic simplified representation of Eq.(123). This is very important in the discussion of consistency problem of the Zeta function equation.

8.2 The problems existing in the Riemann’s deduction of the Zeta function equation

There are many problems in Riemann's original paper in 1859 on the derivation of the Zeta function equation, which the author had discussed in detail in the paper [1]. The most critical one is that the middle term on the right side of Eq.(115) was missing. So Eq.(117) is not equal to zero in general. The calculation result is as follows

                   (128)

If , when, we have, which is the result in the Riemann’s original paper. However, if , when, we have . The right side of Eq.(115) becomes infinite.

Therefore, the integral form of the Riemann Zeta function does not change the divergence of its series summation form. The Riemann Zeta function equation is meaningless in the field . Since the Riemann hypothesis involved the values at the points , the result of Eq. (128) makes the Riemann hypothesis meaningless.

This result also explains why the two sides of Equation (122) of the Riemann Zeta function equation (123) are inconsistent [2]. And why the Riemann hypothesis is so hard to be proved, because the Zeta function equation itself doesn't hold.

In the proof of Eq. (127), Riemann also used a formula of Jacobean function . The formula is also tenable when . If , the formula doesn't make sense. So Eq.(127) is also problematic because the lower limit of integration is in the calculation process, so this formula cannot be used.

In addition, there is another problem that has not been found in the author's paper [1]. Here is a supplement. To derive the formula (111), the order of integral sign and summation sign needs to be exchanged. The condition of interchangeability is that the integrated function is converges uniformly. Since the lower limit of integral is zero, it leads to singularity. The integrated function can not be converges uniformly at point , so the integral sign and the summation sign in Eq. (111) cannot be exchanged. We have

                                  (129)

Therefore, we can not obtain Eq.(113) form Eq.(111).

The key problem is caused by the use of summation formula (112), which is meaningless under conditions . However, the integral lower limit of the Zeta function is zero, which leads to the above problems and makes the Riemann Zeta function equation and the Riemann hypothesis meaningless.  

8.3 The influence of the calculation in this paper on Riemann Zeta function equation

The influences of the calculations in this paper on the Riemann Zeta function equation are below. 

1. According to the calculation in Section 4.3, the single complex continuation of the Gama function is not an analytic function. Therefore, Eqs. (110) and (113) are practically meaningless.

    2. Because Eq.(120) does not hold, we can not get Eqs.(122) and (123). Eq.(121) becomes

                   

                         

                                                (130)

Correspondingly, Eq.(125) becomes

                                             (131)

If we still use the transformation (126), there is no symmetry of formula (127), we have . Because an infinite item is missing, Eq. (131) is still inconsistent and meaningless.

If the missing of the infinite large terms is not taken into account, we suppose that Eq. (131) is correct. According to the understanding of existing theory, the right side of Eq. (131) is regarded as a redefinition of the Riemann Zeta function, then we discuss its nontrivial zero problems.

According to the standard method proposed in the author’s paper [2], the real and the imaginary parts on both sides of Eq.(131) are completely separated, then comparing them. It can be strictly proved with , so the zeros of Eq.(131) are located at the points satisfying . They are ()

                         (132)

No matter whether they are nontrivial zeros or not, the Riemann hypothesis is proved untenable again.  

9.Conclusions

   In this paper, it is proved that the negative continuation of the Gama function in the real number field is still infinite according to the current direct continuation method. The reason is that the form of the Gama function has no any change in the extended field. The single complex continuation of the Gama function does not satisfy the Cauchy-Riemann equation, so it is not an analytic function. But the double complex continuation of the Gama function satisfies the Cauchy-Riemann equation, so it is an analytic function. The complex continuation of the complementary formula of the Gama function is proved to be wrong, and the correct calculation result is given in this paper.

The paper also discusses the complex continuation of the Euler prime formula, and proves that the real part of the formula is different from the original Euler prime formula after the continuation. So it no longer represents the relationship between natural numbers and prime numbers. Therefore, the complex continuation of Euler's prime number formula is meaningless in mathematics. We can not discuss the distribution of prime numbers based on it.

   At the end of this paper, we discuss the influence of the calculation results on the Riemann Zeta function equation and the Riemann hypothesis, and prove again that the Riemann hypothesis does not hold from another angle.

References

1. Mei Xiaochun, The Inconsistency Problem of Riemann Zeta Function Equation, Mathematics Letters, 2019; 5(2): 13-22., doi: 10.11648/j.ml.20190502.11.

2.Mei Xiaochun, A Standard Method to Prove That the Riemann Zeta Function Equation Has No Non-Trivial Zeros, Advances in Pure Mathematics, 10, 8699. https://doi.org/10.4236/apm.2020.102006.

3. Guo, D.R. Method of Mathematics and Physics, High Education Press, Beijing, 1965, P.109.

4. Liang Kunmiao, Mathematics Physics Method, People’s Education Press, 1979, p. 554.

5. Ru Changhai, The Riemann Hypothesis, Qianghua University Publishing Company, 2016, p.61, 52, 192.

6. Bent E. Petersen, Riemann Zeta Function, https://pan.baidu.com/s/1geQsZxL.

7. Mathematics Handbook, Scientific Publishing House, 1980,p. 144.

8. Neukirch, J., Algebraic Number Theory, Springer, Berlin, Heidelberg, 1999 .

9. Fei Liwen, Typical problems and methods in mathematical analysis, High Education Press., 2018,P.563.

Author avatar
Mei Xiaochun
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