THE ZEROSOF HURWITZ'S ZETA-FUNCTION ON ~ = I/2
Steven M. Gonek
Dedicated to Professor Emil Grosswald
I.
Introduction.
Let s = c + i t be a complex variable.
For a fixed a, O < ~ < l , Hurwitz's
zeta-function is defined in the half-plane o > l by
~(s,~) :
~ (n+~)-s,
n=O
and except for a simple pole at s = l , may be a n a l y t i c a l l y continued throughout
the complex plane.
The resemblance of ~(s,~) to Riemann's zeta-function, {(s),
is in certain ways s u p e r f i c i a l .
For besides the two cases ~(s,I/2) = (2S-l)~(s)
and ~(s,l) = ~(s), ~(s,~) possesses neither a functional equation nor an Euler
product.
I t is therefore not surprising that the zeros o f these functions are
distributed d i f f e r e n t l y .
I.
For instance, we note the following:
While ~(s) has no zeros in o > l , ~(s,~) has i n f i n i t e l y many (provided
~ I/2 or l ) .
false.
In p a r t i c u l a r the analogue of the Riemann hypothesis for ~(s,~) is
This was proved by Davenport and Heilbronn [3] when ~ is rational (~ I/2
or l ) or transcendental, and by Cassels I l l when ~ is an algebraic i r r a t i o n a l .
may also prove a quantitative version o f this result [2; p. 1780].
One
Namely, for any
> O, the number of zeros of ~(s,~) (~ ~ I/2 or l ) in the rectangle l < o < I+6,
0 < t < T is : T for s u f f i c i e n t l y large T.
2.
Let o1,o 2 be fixed with I/2 < oI < 02 < I.
Then ~(s,a) has i n f i n i t e l y many
zeros in the s t r i p oI < o < 02 when ~ is rational ( Y 8 9
l ) or transcendental.
The rational case is due to S.M. Voronln [8] (see also S.M. Gonek [ 5 ] ) , the transcendental case to S.M. Gonek [5].
Here too one can show that the number of zeros
up to height T is ~ T for a l l large T.
On the other hand, well-known zero-denslty
estimates imply that ~(s) has at most o(T) zeros in such a rectangle.
130
Pursuing these contrasts f u r t h e r , one might n a t u r a l l y ask whether the l i n e
o = I / 2 is special to {(s,m) as i t
infinity,
We know t h a t as T tends to
the number o f zeros o f e i t h e r f u n c t i o n in the s t r i p 0 < t < T is
T__ log T.
2~
_
is to ~(s).
For ~(s), N. Levinson [7] showed t h a t more than I / 3 o f these zeros
l i e on o = I / 2 ;
i t is w i d e l y held t h a t the correct p r o p o r t i o n is I .
In t h i s paper,
our purpose is to show t h a t f o r c e r t a i n values of m the p r o p o r t i o n of zeros o f
{(s,m) on o = I / 2 is d e f i n i t e l y
less than I.
S p e c i f i c a l l y , we shall prove the
following result.
THEOREM. Let s - ~ ,
~, 4'
1 4'
3 61 or
t h a t the number o f zeros o f ~(s,a)
.
There is a p o s i t i v e constant c < 1 such
(counted according to t h e i r m u l t i p l i c i t i e s )
T
I / 2 + i T ] is ~ (c+o(1)) ~
the segment [ I / 2 ,
on
log T as T tends to i n f i n i t y .
The author would l i k e to take t h i s o p p o r t u n i t y to thank Professor Hugh L.
Montgomery f o r b r i n g i n g t h i s problem to his a t t e n t i o n and Professor Patrick X.
Gallagher f o r p o i n t i n g out an e r r o r in the o r i g i n a l manuscript.
2.
An A u x i l i a r y Lemma.
To prove our theorem we require information about the number of zeros
common to two L - f u n c t i o n s .
due to A. F u j i i
This is provided by the lemma below which is e s s e n t i a l l y
[4; Theorem I ] .
Recall t h a t two D i r i c b l e t characters not induced by the same p r i m i t i v e
character are c a l l e d
inequivalent.
We denote by L(s,•
the D i r i c h l e t L - f u n c t i o n
and x2 are i n e q u i v a l e n t characters.
Let Pl = B1 + iYl denote
with character • .
LEMMA. Suppose
•
a zero of L(S,Xl) w i t h 0 < B1 < I , and w r i t e mi(P I ) f o r the m u l t i p l i c i t y
a zero o f L ( s , x i )
(I)
(i = 1,2).
Then there e x i s t s a p o s i t i v e constant c < 1 such t h a t
Z'
m~n mi(P I ) ~ (c+o(1)) ~ l o g
0 ~ Y1 ~ T I~1,2
as T tends to i n f i n i t y ,
where
o f Pl as
~'
T
means the sum is over d i s t i n c t
zeros
PI"
131
PROOF. We see from the proof of Theorem 1 in F u j i i
primitive
[4; w
that for d i s t i n c t
characters x I , x 2 there exists a p o s i t i v e constant c I < 1 such that as
T tends to i n f i n i t y
(2)
1 > (Cl+O(1)) k
log T.
O<y 1 <T
ml(P I) > m2(P I)
Indeed, (2) holds even when •
as they are inequivalent.
Xl' x2
then •
course i f xi is p r i m i t i v e
But L ( s , •
w
assumes •
' x2
xi = x i ' )
are d i s t i n c t
primitive
and •
L(s,•
as well.
(i = 1,2) and
characters.
have the same modulus.
min
mi(Pl) :
i = 1,2
ml(Pl) -< m2(Pl)
X'
m~(.l)
Hence (2)
However, he l a t e r points out (in
Now
~'
ml(P I) +
0 ~ Y1 ~ T
Z'
ml(Pl ) +
0_< ~i _< T
O<y I <T
(Of
(In the statement of his theorem,
~'
m2(P 1
0 ~ Y1 ~ T
ml(P I) ~ m2(P I)
<
induces •
as long
Therefore (2) is true for the pair L(S,Xl),
that this assumption is unnecessary.)
Z'
0 ~ Y1 ~ T
and x 2 are imprimitive
and L(s,• *i) have the same zeros in 0 < a < I.
is valid for the pair L(S,Xl),
Fujii
or both •
To see t h i s , note that i f •
are inequivalent,
L(s,x~).
•
ml(P I) > m2(P I)
~'
0 _< u _< T
(ml(Pl)-l)
ml(Pl) > m2(Pl)
-
Z'
I.
0_<~ 1 < T
ml(P I) > m2(# I)
The f i r s t
sum on the last line is the total number of zeros of L(s,• I) n
T
0 < o < I, 0 < t < T, and is therefore equal to ( I + o ( I ) ) ~
log T as T tends to
infinity.
Using this and (2) we conclude that
132
~'
0 ~
m~n
ml(P I ) ~ ( I - c i + 0 ( I ) }
~Iog
T.
Y1 ~ T i = 1,2
This establishes (1) with c = 1-c I.
3.
Proof of the Theorem.
For the sake of convenience, we carry out the proof of the Theoremonly
for ~ = I/3 and 2/3.
The modifications required to prove the other cases are
minor and w i l l be discussed at the end of this section.
Throughout we write e(x)
for e2~ix.
We begin with the identity (see Davenport and Heilbronn [3; p. 181])
a = ,__~ X i(a)L(s,x),
~(s, ~)
(3)
x
where 1 < a < q, (a,q) = I , and the sum is over a l l r
q = 3 and assume t h a t a is e i t h e r 1 or 2.
We are then summing over r
characters in (3), both of which are r e a l .
2
3s
characters mod q.
Take
= 2
Thus
a
~(s, ~) : L(s,x0) + x ( a ) L ( s , x ) ,
where x 0 and • are the p r i n c i p a l and nonprincipal characters, r e s p e c t i v e l y , mod 3.
Since L(s,x0) = ( l - 3 - s ) ~ ( s ) ,
the l a s t equation becomes
(4)
2
~
a
~(s, ~) = ( l - 3 - s ) ~ ( s )
REMARK. As w i l l
become apparent, i t
(3) reduce to two terms.
+ x(a)L(s,x).
is essential to our proof t h a t the sum in
This is why the reduced f r a c t i o n ~
have denominator 3,4 or 6.
Now wri te
(5)
~(s) : 89 s(s-l)~-s/2r(~)~Cs)
and
s+l
(6)
~(s,x) :
(~)'T r(~ZlL(s,x).
in the Theorem must
133
Using (5) and (6) to replace
~(s) and L(s,x) in (4) by ~(s) and ~(s,x), and then
s+l
(~) 2 r(_~L) ' we find (after simplifying)
multiplying both sides of (4) by
(7)
~
(3~)-s/2r(~Z)~(s
'
We w r i t e t h i s more b r i e f l y
(a)
~) =/~
(3s/z_3_s/2) r(~s
~
s(s-l)
r(~)
that
{(s)+x(a)~(s,x).
as
A(s)~(s, ~) : B(s)~(s) + x(a)~(s,x),
where
(g)
A(s) = / ~
(3~)-s/Zr(~s
and
(lO)
B(s) : ~/-i-2 (3s/2-3 -s/2)
s(s-l)
r(-~-)
r(~)
Since A(s) never vanishes, the zeros of the right-hand side of (8) are precisely
those of ~(s, ~).
Thus, ~(I/2 + i t O, ~) = 0 i f and only i f the terms on the
right-hand side of (8) cancel or vanish for s = I/2 + i t O.
Since B(s) # 0 on
o : I/2 we see that I/2 + i t 0 is a zero of ~(s, ~) i f and only i f :
I~
II.
~(I/2 + i t O) ~ O, ~(I/2 + ito,•
~ O, and
C(I/2 + ito,•
B(I/2 + i t O) = -x(a) ~(I/2 + i t O)
C(I12 + ito) = ~(I12 + ito,x) : O.
Writing N(T) for the number of zeros (counting m u l t i p l i c i t i e s )
of ~(s, ~) on
[ I / 2 , I/2 + iT] (T > 0), NI(T) for the number of these zeros arising from condition
I , and NIl(T) for the number arising from I I , we see that
(II)
N(T) = NI(T) + NIl(T).
We estimate N(T) by combining estimates for NI(T) and NIl(T).
'
~34
First consider NI(T).
From the relation
~-'~=
~(s,x)
(x
is real) and
the functional equation
~(l-s,x) : ~
where ~(•
:
~(s,•
3
~ x(n)e(~), one easily finds that C(I/2 + i t , x ) is real.
n=l
Similarly ~(I/2 + i t ) is real.
Thus i f t o satisfies I, B(I/2 + i t O) is real.
If
T ~ TO > 0 and i f N~(To,T) denotes the number of solutions of
arg B(I/2 + i t ) ~ 0 (mod ~)
with t e [To,T ] , i t follows that N~(To,T) is an upper bound for the number of
distinct t o ~ [To,T] that satisfy I.
We now prove that there exists a TO such
that N~(To,T) << T for all T ~ TO, and that
{(s, ~) i f
t o satisfies I and t o ~ TO.
I/2 + i t 0 is a simple zero of
These two assertions and the fact that
~(s, ~) has only f i n i t e l y many zeros on [I/2, I/2 + iT O] clearly imply that
(12)
NI(T) << T
(T ~ TO).
To estimate N~(To,T) we examine ~---~arg B(I/2 + i t ) .
for all t since B(s) is analytic and nonzero in 0 < a < I . )
arg B(I/2 + i t ) = arg(
-I
I/4
(The derivative exists
By (I0)
log 3,
) + arg et,t 4~
l e(-t
12~3))
+ a r g ( l - ~33
+ arg(F(43-+ i ~)Ir(l14 + i ~))
or
(13)
arg B(II2 + i t ) :~ +
sin(t log 3)
+ arc t a n ( _
cos(t log 3)
135
where the choice of arguments is immaterial.
The sum of the derivatives of the
f i r s t three terms on the right-hand side of (13) is equal to
log 3
4 - 2 ~ c o s ( t log 3)
Observing that
d
r'
d-~arg r(o+ i t ) = Re ~- (~+ i t )
and using the formula (see Ingham [6; p. 57])
1
r'
~)
9- (s) = log s + o(~--~T
which is valid in larg s[ < ~-a
for any a > O, we find that
d arg(r( 88 i ~)/?(I/4 + i t ) ) < <
d-~
l
t+-TT
for t > O. Thus
d arg B(I/2 + i t ) =
lo 9 3
dt
4-2V'3cos(t log 3)
+ o(tl+-~T)
(t > 0).
-
From this we see that there exists a TO > 0 such that d~arg B(I/2 + i t ) is
bounded and greater than zero for t ~ TO.
That is, arg B(I/2 + i t ) is an increasing
function with bounded derivative on [To,|
Clearly this implies that
N~(To,T) << T
(T ~ TO).
Now suppose that I/2 + i t O is a zero of
{(s, ~) arising from condition I and
that t O ~ TO (TO as above). Differentiating the right-hand side of (8) with
respect to t and evaluating at s = I/2 + i t o, we obtain
(14) ~(I/2 + ito)(~--~)toB(I/2 + i t )
+ B(I/2 + ito)(d~$-)to
~(I/2u~
+ it)+x(a)(~)tD~(I/2u~ + i t , x ) .
136
The second and third terms are real since •
and B(I/2 + i t O) are.
d~ ~(I/2 + i t ) , ~-~ C(I/2 + i t , x )
(Recall that B(I/2 + i t O) is real whenever t O satisfies I . )
If we write B(I/2 + i t ) : IB(I/2 + i t ) l e ( ar9 B(I/2 + i t ) )
2x
the f i r s t term in (14)
becomes
(15)
6(I/2 + ito)e(
arg B(I/2 + i t O)
2~
){(~)toIB(I/2
Since t O satisfies I, e(
arg B(I/2 + ito) )
2~
= •
not equal zero.
Also ( ~ ) t o
chosen), and ~
IB(I/2 + i t ) I
+ it)l+i(~t)to
arg B(I/2 + i t ) } .
and ~(I/2 + i t o ) , which is real, does
arg B(I/2 + i t ) > 0 for t O ~ TO (this is how TO was
is real for all t.
fore (14) have nonvanishing imaginary parts.
I t follows that (15) and there-
Thus I/2 + i t 0 is a simple zero of
the right-hand side of (8) or, what is the same thing, of ~(s, 8).
This f i n a l l y
establishes (12).
We now turn to NIl(T).
Let m(z), ml(z), and m2(z) be the m u l t i p l i c i t i e s of
a
the point z as a zero of ~(s, ~), { ( s ) , and L(s,x) repsectively. By (5), {(s) and
~(s) have the same zeros in 0 < ~ < I ; the same is true for L(s,x) and ~(s,x) in
l i g h t of (6).
Thus t O satisfies
~(s) and L(s,x).
p = 6 + iy
(16)
II i f and only i f I/2 + i t 0 is a common zero of
In p a r t i c u l a r ,
89 i t 0 is a zero of ~(s) on
o = I/2.
Letting
denote a typical zero of ~(s), we then have
NIl(T) :
~'
O<~<T
m(p),
6 = I/2
where as usual ~' means the sum is over d i s t i n c t zeros p .
this we need to consider the numbers m(p).
In order to estimate
From (8) and the fact that B(s) ~ 0
on o = I/2, i t immediately follows that
Ii
min
i=l ,2
mi(I/2 + iy) i f ml(I/2 + iy) ~ m2(I/2 + iy)
m(I/2 + iy)
ml(I/2 + iy) i f ml(I/2 + iy) = m2(I/2 + iy).
137
However, the lower bound this provides for m(I/2 + iy) in the case ml(I/2 + iy) =
m2(I/2 + iy) is of no use to us since we seek an upper bound for NIl(T).
We remedy
this by proving that, except for f i n i t e l y many y, i f ml(ll2 + iy) = m2(I/2 + iy)
then m(I/2 + iy) = ml(I/2 + iy) or ml(I/2 + iy) + l , with the l a t t e r holding at
most O(T) times for y e [O,T].
To show this set ml(I/2 + iy) = m2(I/2 + iy) = k ~ I.
Then the kth derivative
of the right-hand side of (8) with respect to t evaluated at s = I/2 + iy
is
Since the zeros of B(s){(s) + x(a)~(s, X) are those of {(s, ~), we see that
m(I/2 + iy)
> k i f and only i f (17) vanishes.
derivatives of the two
By the definition of k, the kth
~-functions are nonzero at I/2 + iy .
only i f i t s terms cancel.
Hence (17) vanishes
Since x(a), (~t)k ~(I/2 + i t ) , and (~-~)kc(I/2 + i t , x )
are real, this occurs only i f B(I/2 + iy) is real.
B(I/2 + i t ) is real at most O(T) times on [O,T].
But we have already seen that
Thus ml(I/2 + iy) = m2(I/2 + iy)
implies that m(I/2 + iy) = ml(I/2 + iy) ( = k) except for possibly O(T) values of
y e [O,T].
is real).
Suppose now that (17) does vanish at I/2 + iy
(so that B(I/2 + iy)
Taking the k+l st derivative of the right-hand side of (8) with respect
to t and evaluating at s = I/2 + iy , we obtain
) C(I/2 + i t ) ] [ ( ~ ) y B ( I / 2 + i t ) ] + B(I/2 + iy) (d)k+l
~ ~ ~(I12 + i t )
d•
(18) (k+l)[(
+ x(a)(d~)~+l{(I/2 + i t , x ) .
As in our analysis of (14), we find that the second and third terms are real and
that the f i r s t has nonvanishing imaginary part when y is large.
nonzero and m(1/2 + iy) = k+l = ml(I/2 + iy) + l (for large
y).
To summarize: there exists a TO > 0 such that i f I/2 + iy
~(s) with y
Thus (18) is
is a zero of
> TO, then
m(I/2 + iy) = min mi(I/2 + iy) or min mi(I/2 + iy) + I ;
i=1,2
i:I,2
138
the second case occurs at most O(T) times on [To,T].
We can now bound N I l ( T ) .
NIl(T) =
Writing (16) as
~'
m(p) + 0 ( I )
TO_<Y<T
6
= I/2
and using the previous r e s u l t , we have
NIl(T) =
Z'
TO _< y <_ T
mi(P) + O(T)
min
i = 1,2
B = I/2
Z'
O<y<T
min
i=1,2
mi(P) + O(T)
6 : I/2
~'
O<~<T
min
i=1,2
mi(P) + O(T),
where the f i n a l sum is over the d i s t i n c t zeros p of ~(s) with 0 < 6 < I ,
0 < ~ < T.
Applying the Lemma to the l a s t sum (note t h a t ~(s) is an L - f u n c t i o n )
we see t h a t as T tends to i n f i n i t y
(19)
N I l ( T ) ~ (c+o(1)) ~--~-log T,
where c is a p o s i t i v e constant
The proof o f the Theorem
< I.
f o r ~ = I / 3 and 2/3 now f o l l o w s from ( I I ) ,
(12),
and (19).
Our proof c a r r i e s over to the cases
s l i g h t changes in the formulae.
corresponding to (8),
~ = I/4,
3/4. I / 6 , and 5/6 w i t h only
For instance, i f ~ = a/4, a = 1 or 3, then
(9), and (I0) we have
a
A(s)~(s, ~) = B(s)~(s) + x ( a ) ~ ( s , x ) ,
A(s) : 4__ ( 4 ~ ) - s / 2 r ( sz + l ) ,
and
139
where x is the nonprincipal character mod 4.
a
When ~ = ~ , a = 1 or 5, the s i t u a t i o n is only s l i g h t l y more complicated.
The nonprincipal c h a r a c t e r x mod 6 is induced by the p r i m i t i v e c h a r a c t e r x mod 3.
Also, for the p r i n c i p a l c h a r a c t e r
Xo mod 6 we have L(S,•
= (1-2-s)(1-3-s)~(s).
Thus, in place o f (4) we obtain
2
a
6~ ~(s, ~)
: ( l - 2 - S ) ( l - 3 - s ) ~ ( s ) + x * ( a ) ( l + 2 - S ) L ( s , x ) *,
and instead of (8),
(9),
(I0) we have
a
*
*
A(S)~(S, ~-) : B(s)c(s) + X (a)c(s,•
A(s) : ~ 2
(l,2x)-s/2
r(_~]_),
(l+2-s)
and
B(s) :
In e i t h e r case A(s) ~ 0 f o r
(3s/z-3-s/2)II-z-S)
s(s-l)(l+2 -s)
0 < o <
1 and ~ a r g
r(~)
1
B(~ + i t ) i s
bounded and
0
>
f o r a l l large t.
4.
A Conjecture.
We expect the Lemma, and t h e r e f o r e the Theorem, to be f a r from best possible.
Indeed, i t
is g e n e r a l l y held t h a t no two L-functions with i n e q u i v a l e n t characters
have common zeros in 0 < ~ < I .
On t h i s assumption we would have N I l ( T ) << T
instead of (19) and t h i s along w i t h ( I I )
and (12) implies t h a t N(T) << T.
I t is
p l a u s i b l e to suppose t h a t these bounds are v a l i d f o r o t h e r r a t i o n a l values o f
so we make the f o l l o w i n g
CONJECTURE.
zeros on [ I / 2 ,
I f e is r a t i o n a l , 0 < ~ <
I/2 + iT].
I , and
~ ~ I / 2 , then ~(s,~) has <<
T
140
REFERENCES
I.
J.W.S. Cassels, Footnote to a note of Davenport and Heilbronn, J. London Math.
Soc. 36 (1961), 177-184.
2.
H. Davenport, The collected works
of Harold Davenport, vol. 4, Academic Press,
New York, 1977.
3.
H. Davenport and H. Heilbronn, On the zeros of certain D i r i c h l e t series, I.
J. London Math. Soc. I I (1936), 181-185.
4.
A. F u j i i , On the zeros of D i r i c h l e t L-functions (V), Acta A r i t h . 28 (1976),
395-403.
5.
S.M. Gonek, A n a l y t i c properties of zeta and L-functions, Thesis, U n i v e r s i t y
of Michigan, 1979.
6.
A . E . Ingham, The distribution of prime numbers, Cambridge University Press,
London, 1932.
7.
N. Levinson, More than one-third of the zeros of Riemann's zeta-function are
on ~ = I / 2 , Advances in Math. 13 (1974), 383-436.
8.
S.M. Voronin, On the zeros of zeta-functions of quadratic forms, Trudy Mat.
Inst. Steklov 142 (1976), 135-147. See also: Proc. Steklov Inst. Math. 3
(1979), 143-155.
Department of Mathematics
U n i v e r s i t y of Rochester
Rochester, NY 14627