Journal of Number Theory NT2137
journal of number theory 65, 325333 (1997)
article no. NT972137
The Positivity of a Sequence of Numbers and
the Riemann Hypothesis
Xian-Jin Li
The University of Texas at Austin, Austin, Texas 78712
Communicated by A. Granville
Received September 10, 1996; revised December 16, 1996
In this note, we prove that the Riemann hypothesis for the Dedekind zeta
function is equivalent to the nonnegativity of a sequence of real numbers.
1997 Academic Press
1. THE RIEMANN ZETA FUNCTION
Let [* n ] be a sequence of numbers given by
(n&1)! * n =
d n n&1
[s
log !(s)] s=1
ds n
(1.1)
for all positive integers n, where
!(s)=s(s&1) ? &s21
s
\2+ `(s)
with `(s) being the Riemann zeta function.
Theorem 1. A necessary and sufficient condition for the nontrivial zeros
of the Riemann zeta function to lie on the critical line is that * n is nonnegative for every positive integer n.
Proof.
.(z)=!
Define
1
=4
1&z
\ +
|
[x 32$(x)]$ (x &12x 12(1&z) +x &12(1&z) ) dx
(1.2)
1
325
0022-314X97 25.00
Copyright 1997 by Academic Press
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XIAN-JIN LI
for z in the unit disk, where
2
(x)= : e &?n x.
n=1
Write
\
s
\ \+
!(s)=` 1&
(1.3)
where the product is taken over all nontrivial zeros of the Riemann zeta
function with \ and 1&\ being paired together. It follows that
.(z)=`
\
1&(1&(1\)) z
.
1&z
A necessary and sufficient condition for the nontrivial zeros of the Riemann
zeta function to lie on the critical line is that .$(z).(z) is analytic in the
unit disk. Put
.$(z)
= : * n+1 z n
.(z) n=0
for |z| < 14 , where
\
1
\
n
_ \ +&
* n =: 1& 1&
(1.4)
for every positive integer n. On the other hand, by (1.3) we have
n&1
1
d n n&1
n
[s
log
!(s)]
=&:
:
(\&1) k&n
s=1
n
(n&1)! ds
k
\ k=0
\+
\
1
\
n
_ \ + &,
=: 1& 1&
and hence * n is also given by the expression (1.1). Let
.(z)=1+ : a j z j.
(1.5)
j=1
We find that
n
(&1) l&1
l
l=1
* n =n :
:
a k1 } } } a kl
1k1 , ..., kl n
k1 + } } } +kl =n
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327
THE RIEMANN HYPOTHESIS
for every positive integer n. Expanding the right side of (1.2) in power
series (1.5), we find that
( j+n) } } } ( j+1)
n! (n+1)! 2 n
n=0
a j =2 :
|
[x 32$(x)]$ (x &12 +(&1) n+1 )(log x) n+1 dx
1
(1.6)
for every positive integer j. By (1.6) we can write
(n+j ) } } } (n+1)
j! (n+1)! 2 n+1
n=0
a j =4 :
=
|
[x 32$(x)]$ (x &12 +(&1) n+1 )(log x) n+1 dx
1
4 d j j&1 (t2) n+1
t
:
j! dt j
n=0 (n+1)!
{
|
[x 32$(x)]$
1
_(x &12 +(&1) n+1 )(log x) n+1 dx
j
4 d
t j&1
j! dt j
=
t=1
{ [x $(x)]$ (x [e &1]+[e
4 d
=
t
+e
)
| [x $(x)]$ (x e
=
j! dt {
j&1 1
1
=4 :
[x $(x)]$
|
\ j&l + l!
\2 log x+ [1+(&1) x
=
j
j&1
j
|
32
&12
(t2) ln x
&(t2) ln x
&1])
1
&12 (t2) ln x
32
t=1
&(t2) ln x
1
j
=
t=1
l
l
32
&12
] dx.
1
l=1
This expression implies that a j is a positive real number for every positive
integer j. Since the identity
: na n z n&1 =
n=1
\
: ai zi
i=0
+\
: * j+1 z j
j=0
+
holds, we have the recurrence relation
n&1
* n =na n & : * j a n&j
j=1
for every positive integer n.
By (1.1), * n is a real number for every positive integer n. If the nontrivial
zeros of `(s) lie on the critical line, then |1&(1\)| =1 for every nontrivial
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XIAN-JIN LI
zero \ of `(s). Put 1&(1\)=exp(i% \ ) for some real number % \ . Then by
(1.4) we have
* n =: (1&e in%\ )=: (1&cos n% \ ).
\
\
This implies that the number * n is nonnegative for every positive integer n.
Conversely, if the number * n is nonnegative for every positive integer n,
then
* n na n
for every positive integer n. It follows that
: |* n z n&1 | : na n |z| n&1 =.$( |z| )<
n=1
n=1
for z in the unit disk. This implies that .$(z).(z) is analytic in the unit
disk.
This completes the proof of the theorem.
2. THE DEDEKIND ZETA FUNCTION
Let k be an algebraic number field with r 1 real places and r 2 imaginary
places. The Dedekind zeta function ` k(s) of k is defined by
` k(s)=` (1&Np &s ) &1
p
for Re s>1, where the product is taken over all the finite prime divisors
of k. Put G 1(s)=? &s2 1(s2) and G 2(s)=(2?) 1&s 1(s). Define
Z k(s)=G 1(s) r1 G 2(s) r2 ` k(s).
By Theorem 3 of Chapter VII, Section 6, of [4], the function Z k(s) is
analytic in the complex plane except for simple poles at s=0 and s=1, and
satisfies the functional identity
Z k(s)= |d| (12)&s Z k(1&s)
where d is the discriminant of k. Its residues at s=0 and s=1 are respectively &c k and |d| &12 c k with c k =2 r1 (2?) r2 hRe, where h, R, and e are
respectively the number of ideal classes of k, the regulator of k, and the
s2
Z k(s). Then
number of roots of unity in k. Let ! k(s)=c &1
k s(s&1) |d|
! k(s) is an entire function and ! k(0)=1.
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THE RIEMANN HYPOTHESIS
329
Let [* n ] be a sequence of numbers given by
(n&1)! * n =
d n n&1
[s
log ! k(s)] s=1
ds n
for all positive integers n. The aim now is to prove the following theorem.
Theorem 2. A necessary and sufficient condition for the nontrivial zeros
of the Dedekind zeta function ` k(s) to lie on the critical line is that * n is nonnegative for every positive integer n.
3. PROOF OF THE THEOREM 2
Lemma 3.1.
The identity
\
1
\
n
\ \ ++
* n =: 1& 1&
holds for every positive integer n, where summation is taken over all nontrivial zeros of the Dedekind zeta function ` k(s) with \ and 1&\ being paired
together.
Proof.
of [2])
By Theorem 2 of Barner [1], we have the formula (cf. Chapter 2
s
,
\
\ +
! k(s)=` 1&
\
(3.1)
where the product is taken over all zeros of ! k(s) with \ and 1&\ being
always paired together. An argument similar to that made for the Riemann
zeta function in Chapter 2 of [2] shows that the convergence of the
product (3.1) is uniform on compact subsets of the complex plane.
Since ! k(s)=! k(1&s), we have
d n n&1
dn
n
[s
log
!
(s)]
=(&1)
[(1&s) n&1 log ! k(s)] s=0 . (3.2)
k
s=1
ds n
ds n
Since ` k(s) does not vanish at s=0, we can write
\ &m m
s
m
m=1
log ! k(s)=&: :
\
(3.3)
where |s| <= for a sufficiently small positive number =, where \ and 1&\
are paired together in the summation over \. Since the product (3.1)
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XIAN-JIN LI
converges uniformly, the series (3.3) converges uniformly for |s| <=. It
follows that
n
dn
n
1
n&1
[(1&s)
log
!
(s)]
=&:
:
(&1) n&m
\ &m.
k
s=0
n
(n&1)! ds
m
\ m=1
\+
This formula together with (3.2) implies the stated identity. K
Define
.(z)=! k
1
1&z
\ +
for z in the unit disk. Since the function ! k(s) is analytic in the complex
plane of s, the function .(z) is analytic in the unit disk.
Lemma 3.2.
Let
.(z)=1+ : a j z j.
j=1
Then the coefficient a j is a positive real number for every positive integer j.
Proof. Define = v to be one when v is a real place of k and to be two
when v is an imaginary place of k. Let x=> x v be the variable in the half
space R r+1 +r2. Denote by |x| the product > x =vv , which is taken over all
infinite places of k. If N=r 1 +2r 2 , then the Hecke theta function 3 k(x) is
defined by
\
3 k(x)=: exp &? |d| &1N (Nb) 2N : = v x v
b
v
+
where the summation over b is taken over all nonzero integral ideals of k
and where the summation over v is taken over all infinite places of k. Put
dx=> dx v . It follows from Theorem 3 of Chapter XIII, Section 3, in [3]
that
! k(s)=1+c &1
k s(s&1)
|
3 k(x)( |x| s2 + |x| (1&s)2 )
|x| 1
dx
.
x
(3.4)
Let
|
3 k(x)( |x| 12(1&z) + |x| 12 |x| &12(1&z) )
|x| 1
dx
= : b m z m.
x m=0
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(3.5)
331
THE RIEMANN HYPOTHESIS
It is clear that b 0 is a positive number. We have
(m+n) } } } (m+1)
n! (n+1)! 2 n+1
n=0
bm = :
|
3 k(x)(1+ |x| 12 (&1) n+1 )(log |x|) n+1
|x| 1
dx
x
for every positive integer m. By computation, we find that
bm=
1 (n+m) } } } (n+1)
:
m! n=0 (n+1)! 2 n+1
|
3 k(x)(1+ |x| 12 (&1) n+1 )(log |x| ) n+1
_
=
|x| 1
1 d m m&1
t
m! dt m
\
|
dx
x
3 k(x)(e (t2) log |x| + |x| 12 e &(t2) log |x| )
|x| 1
dx
x
+
.
t=1
It follows that
m
bm= :
l=1
\
m&1 1
m&l l !
+
|
3 k(x)
|x| 1
\
1
log |x|
2
l
+ ( |x|
12
+(&1) l )
dx
x
(3.6)
for every positive integer m. Since 3 k(x) is positive for every x in R r+1 +r2 ,
it follows from (3.6) that the coefficients b m are positive real numbers for
all nonnegative integers m.
The identity
z
q
2 = : qz
(1&z)
q=1
holds for z in the unit disk. It follows from (3.4) and (3.5) that
j&1
c k a j = : ( j&m) b m
(3.7)
m=0
for every positive integer j. Since b m are positive numbers for all nonnegative integers m, we see that a j is a positive real number for every
positive integer j. K
Proof of the Theorem. Since ! k(1)=1 and ! k(s)=! k(1&s), it follows
from the product formula (3.1) that
.(z)=`
\
1&(1&(1\)) z
.
1&z
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(3.8)
332
XIAN-JIN LI
Since ! k(s) does not vanish at s=1, we can write
.$(z).(z)= : * n+1 z n
(3.9)
n=0
by using the formula (3.8) when |z| <= for a sufficiently small positive
number =. Since
: na n z n&1 =
n=1
\
: ai zi
i=0
+\
+
: * j+1 z j ,
j=0
we have
n&1
* n =na n & : * j a n&j
(3.10)
j=1
for n=2, 3, ..., where * 1 =a 1 and a 0 =1.
If the nontrivial zeros of ` k(s) lie on the critical line, it follows from
Lemma 3.1 that the numbers * n are nonnegative for all positive integers n.
Conversely, assume that the number * n is nonnegative for every positive
integer n. It follows from (3.10) and Lemma 3.2 that
* n na n
for every positive integer n. This inequality together with Lemma 3.2
implies that
: |* n z n&1 | : na n |z| n&1 =.$( |z| )
n=1
(3.11)
n=1
for z in the unit disk. Since .$(z) is analytic in the unit disk, .$(|z| ) is finite
for z in the unit disk. It follows from (3.9) and (3.11) that .$(z).(z) is
analytic in the unit disk. It is clear that a necessary and sufficient condition
for the nontrivial zeros of the Dedekind zeta function ` k(s) to lie on the
critical line is that .$(z).(z) is analytic in the unit disk. Therefore, the
nontrivial zeros of the Dedekind zeta function ` k(s) lie on the critical line.
This completes the proof of the theorem. K
Remark. We know from the proof of Theorem 2 that * 1 =a 1 , which is
a positive number by Lemma 3.2. An explicit expression for * n is implicit
in the recurrence relation (3.10) together with formulas (3.6) and (3.7).
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THE RIEMANN HYPOTHESIS
333
ACKNOWLEDGMENT
The author thanks the referee for valuable suggestions on improving the presentation of the
original version of this paper.
REFERENCES
1.
2.
3.
4.
K. Barner, On A. Weil's explicit formula, J. Reine Angew. Math. 323 (1981), 139152.
H. M. Edwards, ``Riemann's Zeta Function,'' Academic Press, New York, 1974.
S. Lang, ``Algebraic Number Theory,'' AddisonWesley, Reading, MA, 1970.
A. Weil, ``Basic Number Theory,'' Springer-Verlag, Heidelberg, 1967.
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