A Necessary Condition for the Existence of the
Nontrivial Zeros of the Riemann Zeta Function
Fayang G. QIU
Nanfang Laboratory of Molecular Engineering 190 Kaiyuan Avenue, The Science Park of Guangzhou, Guangdong 510530, PR China
Keywords: Nontrivial zeros, Gamma function, Reflection functional equation, Zeta function, riemann Hypothesis.
Abstract. Starting from the symmetrical reflection functional equation of the zeta function, we have found that the values satisfying (s) = 0 must also satisfy (s) = (1 - s), from which we have
shown that all nontrivial zeros of the zeta function must have = ½. MSC numbers: 11M26, 11M41
Introduction
The Riemann zeta function was originally expressed by the following series:
...
3
1
2
1
1
1
)
(
1
s s
n s
n
s
(1)which converges and defines a function analytic in the region {s in C: Re(s) > 1}. Since the series diverges in the other region {s in C: Re(s) ≤ 1}, Eq. 1 is no longer valid. Riemann realized that the zeta function can be redefined using an analytic continuation and he extended it in a unique way to a meromorphic zeta function (s) defined for all complex numbers s except s = 1, where the zeta function has a singularity.
Riemann has established[1] that the zeta function satisfies the functional equation
)
1
(
)
1
(
2
sin
2
)
(
s
s s 1s
s
s
(2)where Γ(s) is the gamma function valid on the entire complex plane. Owing to the properties of the sine function, the functional equation implies that ζ(s) has simple zeroes at negative even integers, which are known as the trivial zeros of ζ(s). In addition to these trivial zeros, the extended zeta function has many nontrivial zeros in the strip {s∈C: 0 < Re(s) < 1}, which is called the critical strip. The nontrivial zeros are more important not only because their distribution is far less understood, but their studies have yielded important results concerning prime numbers and related subjects in number theory.[2,3] When Riemann calculated a few nontrivial zeros, it appeared that all had R [s]=½. Thus he hypothesized that all nontrivial zeros of the zeta function have real part R [s] = ½, which is confirmed to be true for the first 1012 roots.[2] In the theory of the Riemann zeta function, the set {s∈C: Re(s) = 1/2} is called the critical line.
An equivalent relationship had been conjectured by Euler over a hundred years earlier, in 1749, for the Dirichlet eta function (also known as the alternating zeta function), [2]
)
(
)
2
1
(
)
1
(
)
(
11
1
s
n
s
sn
s n
(3))
(
)
2
(
)
1
(
2
1
)
(
s
s/2s
s
s
s
(4)
and then the functional equation is given by
)
1
(
)
(
s
s
(5))
1
(
)
2
1
(
)
(
)
2
(
21
2
s
s
s
s
s s
(6)In the past one hundred and fifty years, the intense research around the distribution of the nontrivial zeros of the zeta function have resulted in great progress both in the development of new mathematical tools and new insights into the zeta functions. However, at present, the Riemann Hypothesis still remains to be proven. Since the Riemann Hypothesis asserts that all nontrivial zeros are on the critical line, it is equivalent to the statement that R [s] = ½ is a necessary condition for the zeta function to have nontrivial zeros.
The Necessary Condition
Taking the absolute value for both sides of Eq. 6, one obtains
|
)
1
(
||
)
2
1
(
|
|
)
(
||
)
2
(
|
21
2
s
s
s
s
s s
(7) Since neither the exponential nor the gamma function has a zero, it is obvious that both s/2(s/2) > 0 and (1s)/2((1s)/2) > 0 in the critical strip. If (s) ≠ (1 - s), then, at least
one of them is nonzero. However, according to Eq. 7, if (s)> 0, then (1 - s) > 0, and vice versa. In this case, neither (s) nor (1 - s) can be zero. Hence no nontrivial zeros of (s) will occur in the area where (s) ≠ (1 - s). Thus, all nontrivial zeros will be bound to the values that satisfy
Eq. 8:
)
1
(
)
(
s
s
(8) For the sake of convenience, s is replaced with ( + ib) in the following discussion.
While Eq. 7 is good for all values in the critical strip, only those values that may contain nontrivial zeros will satisfy both Eq. 7 and Eq. 8. Thus, after Eq. 7 and Eq. 8 are combined, one can easily realize that only those values that may contain nontrivial zeros will satisfy Eq. 9:
0
|
)
(
|
|
)
2
1
(
|
|
)
2
(
|
21
2
ib
ib
ib
ibib
(9) where ( + ib)≥ 0, and we will respectively treat the ( + ib)= 0 and the ( + ib)> 0 cases
to solve Eq. 9 for the values.
If ( + ib)= 0, which may happen incidentally for a certain set of (, b) value, then we immediately know the value at which the zeta function has a nontrivial zero. No further discussion is necessary for such a case. In fact, it is impossible to obtain an exact nontrivial zero since all the nontrivial zeros have irrational imaginary part.
|
)
2
1
(
|
|
)
2
(
|
21
2
ib
ib
ib
ib
(10)
In this case, Eq. 8 and Eq. 10 are mutually sufficient and necessary conditions for each other. We will see what values will satisfy Eq. 10.
The complex gamma function defined in the Weierstrass form is: [4]
e
e
z nn z
n
z
z
z
/1 1
1
)
(
(11)
The absolute value of the gamma function can thus be obtained:
e
e
z nn z
n
z
z
z
/1 1
1
)
(
(12)
where is the Euler-Mascheroni constant. After replacing z with (+ ib)/2, one finds
e
e
e
e
n n
n ib
n ib ib
n
b
n
b
n
ib
ib
2 2 2
1 2 2 2
2 1
2 2
1
2
)
2
(
2
2
1
2
)
(
(13) Thus, Eq. 14 and Eq. 15 are obvious:
e
e
nn ib
n
b
n
b
ib
2 2 2
1 2 2 2 2 1
2
2
)
2
(
2
|
)
2
(
|
(14)
e
e
nn ib
n b n
b ib
2 1 2 2 1
2 1 2 2 2
1 1
2 1
2 ) 1 2 ( 2
) 1 ( |
) 2 1 ( |
(15) According to the reciprocals of both sides of Eq. 10, it is obvious that the left hand side of Eq. 14 equals that of Eq. 15. If (+ ib) = 0, then Eq. 14 is zero while Eq. 15 is not; the opposite is true if (1 - - ib) = 0. Thus, no nontrivial zeros will occur when either (+ ib) = 0 or (1 - - ib) = 0, and these two points i.e., (0, 0) and (1, 0), will be excluded in the following discussion.
Dividing (15) with (14):
1
)
2
(
)
1
2
(
)
1
(
2 2 1 2 2
2 2
1 2 2
2 2 2
2 1 2
2 1
e
e
nn
n
b
b
n
b
b
(16) and then taking the natural logarithm, one finds that
0 )
2 (
) 1 2 ( ln 2 1 2
2 1 )
1 ( ln 2 1 2
2 1 ln 2
2 1
2 2
2 2 1
2 2
2 2
nn bb nb b
n
determine if the total sum of the two series is convergent. The second series may be expanded as follows:
2 2 1 2 2 2 1 2 2 2 2 2 1 2 2 2 2 1 2 2 2 2 1 ) 2 ( ) 2 1 )( 1 4 ( 1 ln 2 1 ) 2 ( ) 2 1 ( ) 2 1 )( 2 ( 2 1 ln 2 1 ) 2 ( ) 2 1 ( ) 2 1 )( 2 ( 2 ) 2 ( ln 2 1 ) 2 ( ) 2 1 2 ( ln 2 1 ) 2 ( ) 1 2 ( ln 2 1 b n n b n n b n b n n b n b n b n b n n n n n n (18) For any x > 0，since 1/(1+x) < ln(1+1/x) < 1/x, thus, for ln(1+1/x) < 1/x, we have
2 2 12 2
1 (2 )
1 4 2 2 1 ) 2 ( ) 2 1 )( 1 4 ( 1 ln 2 1 b n n b n n n n (19) When the up limit of the sum of the natural logarithm series is used, the low limit of the total sum of the two series is
2 2 2 2 1 2 2 2 2 2 1 2 2 1 2 2 1 ) 2 ( 4 2 2 1 ) 2 ( 4 ) 2 ( 2 2 1 ) 2 ( 1 4 1 2 2 1 ) 2 ( ) 2 1 )( 1 4 ( 1 ln 2 1 2 2 1 nb n n n b n nb n n n n b n b n n n b n n n n n n n (20) which is convergent for any pair of (, b). On the other hand, since ln(1+1/x) > 1/(1+x), we have) 2 1 )( 1 4 ( ) 2 ( ) 1 4 ( 2 2 1 ) 2 ( ) 2 1 )( 1 4 ( 1 ln 2 1 2 2 1 2 2
1
n b n n b n n n n (21) The up limit of the total sum of the two series is given as follows:
)) 2 1 )( 1 4 ( ) 2 (( 2 1 4 3 2 2 1 )) 2 1 )( 1 4 ( ) 2 (( 2 8 1 4 4 2 2 1 )) 2 1 )( 1 4 ( ) 2 (( ) 1 4 ( ) 2 1 )( 1 4 ( ) 2 ( 2 2 1 ) 2 1 )( 1 4 ( ) 2 ( 1 4 1 2 2 1 ) 2 ( ) 2 1 )( 1 4 ( 1 ln 2 1 2 2 1 2 2 2 2 1 2 2 2 2 1 2 2 2 2 1 2 2 1 2 2 1 n b n n b n n n b n n n n n b n n b n n n n n b n n b n n n b n n n n n n n n (22) which is also convergent for any set of (, b) in the critical strip. Therefore, total the sum of the two series is bound by a low and an up limit defined by Eq. 20 and Eq. 22, respectively. However, for a given , the sum of the terms containing b and the sum of the rest terms that are independent of bC b
n
b n
n b
b n
-)
2 (
) 1 2 ( ln 2 1 2
2 1 )
1 ( ln 2 1
2 2
2 2 1
2 2
2 2
(23)
C
2 2 1 ln 2
2
1 (24)
where C is a constant. In Eq. 23, for any given , since C will change as b changes for any value
other than = 1 - , or, 1 -2 = 0, while the numeric value of C is independent of b in Eq. 24. This can be easily verified by choosing a few sets of special values of (, b) with Eq. 23. Thus, the change must be cancelled out, which can be realized iff (1-2) = 0. Therefore, = ½ is the only numeric solution to both Eq. 23 and Eq. 24.
Thus, = ½ is the only numeric solution to Eq. 17. Consequently, all nontrivial zeros of the zeta functions must have real part = ½. Since this conclusion is solely based on the reflection functional equation and is independent of the form that the zeta function may take, all the nontrivial zeros of the general L-functions must also have real part = ½.
Summary
Starting from the symmetrical reflection functional equation of the zeta function, we have found that the values satisfying (s) = 0 must also satisfy (s) = (1 - s). This is a necessary condition
for the existence of the nontrivial zeros. We have shown that = ½ is the only numeric solution that satisfies this requirement.
Acknowledgement
This research was financially supported by Nanfang Laboratory of Molecular Engineering, Guangzhou, 510530, China.
References
[1] Riemann, G. F. B. Über die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsber. Königl. Preuss. Akad. Wiss. Berlin, 671-680, Nov. 1859.
