http://dx.doi.org/10.4236/apm.2013.39098
Value Distribution of L-Functions with Rational Moving
Targets
Matthew Cardwell1, Zhuan Ye2* 1Intelligent Medical Objects, Inc., Northbrook, USA
2Department of Mathematical Sciences, Northern Illinois University, DeKalb, USA Email: mcardwell@e-imo.com, *ye@math.niu.edu
Received August 26, 2013; revised September 26, 2013; accepted October 1,2013
Copyright © 2013 Matthew Cardwell, Zhuan Ye. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In accordance of the Creative Commons Attribution License all Copyrights © 2013 are reserved for SCIRP and the owner of the intellectual property Matthew Cardwell, Zhuan Ye. All Copyright © 2013 are guarded by law and by SCIRP as a guardian.
ABSTRACT
We prove some value-distribution results for a class of L-functions with rational moving targets. The class contains Selberg class, as well as the Riemann-zeta function.
Keywords: Value Distribution; Moving Target; L-Function; Selberg Class
1. Introduction
We define the class to be the collection of functions
1
s,n
L s
a n n satisfying Ramanujan hypothesis,Analytic continuation and Functional equation. We also denote the degree of a function L by dL which is a non-negative real number. We refer the reader to Chapter six of [1] for a complete definitions. Obviously,
the class contains the Selberg class. Also every
function in the class is an -function and the
Riemann-zeta function is in the class. In this paper, we prove a value-distribution theorem for the class with rational moving targets. The theorem generalizes the value-distribution results in Chapter seven of [1] from fixed targets to moving targets.
L
Theorem. Assume that L
1and is a rational
function with . Let the roots of the
equation be denoted by
R
limsR s
0R s
L s R RiR.
Then
(I) For any > max 1,1 1 2 L
b
d
,
2
, as .
R R
R b T T
b O T T
(II) For sufficiently large negative b,
< 2
4
2π log log ,
as .
R
R L
T T
T
b b d T O T
e T
Proof of (I). It is known that if L, then
01
1 , as
s n
a n
L s O k
n
;
where 0 is the index of the first non-zero term of the
sequence of k
a n
n 2 , s it with ,t. Since
R s
0limL s , there exists 0> 0 such that
L s R s 0 for Re >s 0 . It follows that
0
<
R
for all real part of zeros of the function
L s R s . We set R z
P
q
z
,
Q z where the
degrees of P Q, are p , respectively; and define
s s R s
.
Thus, there is 1 such that is analytic in the
region
> 1
r
1
>
s r since is a meromorphic function in
with the only pole at . We apply Littlewood’s
argument principle [3] to in the rectangle
L
s1
c
: c T, 2
1,it b
t T
> max c
where are
parameters satisfying . Thus,
, c T 0 b T
, >r1
log d 2π , d
b
s s i
where the given logarithm is defined as in Littlewood’s argument principle [3]. To prove our result, however, we first decompose our auxiliary function by
1
2
1 : for
1 : for >
L s
P s P s s p q
P s Q s
s
L s
R s R s s p q
R s
1
(1)
Without loss of generality, we may assume that
whenever since we can always write
,
p q pq
N
N
R s P s s Q s s for due to our
choice of the parameters which define the rectangle . However, the modification will guarantee in the case of
0 s
1
k that exhibit polynomial growth, which is
necessary for our proof. In the case of ,
already exhibits polynomial growth, and no such adjustment is necessary. We now integrate the logarithm of to get
,
P Q
>
p q R
1
2
log log d for log d
log log d for >
s P s s O T p q
s s
s R s s O T p q
where the terms are the integrals of the maxi-
mum contribution from writing
O T
s as a sum of loga-
rithms. By our choice of T , both and
are analytic in Hence, Cauchy’s Theorem gives
logP logR
.
1
2
log for log
log f
d d
or d
s s O T p q
s s
s s O T p q
(2)
To connect this integral with Littlewood’s argument principle [3], we note that the definition of guaran- c
tees that
2
2
d d
2π , 2π
2π .
R R
R R
R
c
b b
b T T
R b T T
i i
i b
(3)
In light of (2) and because the quantity given in (3) is imaginary-valued, we get for k1, 2
2
2
2
2 2
2π
Im log arg log arg
log 2 arg 2 log arg
log log arg a
d d
d d
d d d rg
R R
R b T T
c T
k k k k
b T
c T
k k k k
b T
T T c c
k k k
T T b b
i b
i iT i iT i c it i c it t
iT i iT i b it i b it t O T
i b it t c it t iT
4, 1
2
:
d
,
k
j k j
iT O T
I O T
1
(4)
for instance.
We now estimate I1,k. For T large enough, we have for tT k, (since p q, 1),
1
1 1
log log log
log 1 log 1 log log 2.
L b it L b it
b it
P b it Q b it P b it Q b it
L b it L b it L b it
Then for large enough, , we find in a
similar fashion that
T tT k, 2
2
log log 1
log log 2.
L b it b it
R b it
b it
Since we have the same estimate for , we find
that
1, 2
k
2
1, 1,
2 2
2 2
, log
log 2
1 log 2
d
d
d
T
k k T
T T
T T
I T b I b it t O T
b it T
t O T
T
T
L b it t O T
T
It is known [2] that for > max 1,1 1 2 L
b
d
,
2
2 2
2 1
d
1 1 .
lim TT
T n
a n
L b it t O
T n
Hence, I1,k
T b,
O T
uniformly in1 1 ,1 2 dL
> max
b .
We next move to estimate 2,k. For sufficiently large
positive real number , we have I c
1 and
1,L c it L c it
P c it R c it
(5)
so
1
log c it log 1 L c it
P c it
since q1. Furthermore,
2
log c it 1 L c it .
R c it
Since we may take large enough so that c
1k cit
, we may write using a
Taylor series expansion in the rectangle log k
. For c it
k1, we have after taking real parts that
1
1
1 1
1
1 1 1 1
1
log Re
1
Re .
k
k k c it
k n
k
k c
k n n
k
a n c it
n k P c it
a n a n
n n
k P c it
it
We now observe that for sufficiently large T and some constant M we have
2 1
1
1,
T k
k it k
T
k
t T
MT
P c it n n P c iT
for kN and
1 1
1 1 1
1 lim sup
k k
c c
k k n n n n
for sufficiently large . In light of these bounds and the definition of , we have (6)
c
where the last equality holds because c could be
sufficiently large. Replacing by in the above
computations, we see analogously that
P R
2,2 1
I O .
Finally, we estimate I3,k and I4,k. We show the com-
putation for I3,k explicitly and note that the bound for
follows analogously. We first suppose that
4,k
I k
iT
N zeros for
has exactly b c. T are at
most N 1
hen, there
subinterv ounting for multiplicities, in
which
als, c
Re k iT is of consta signt n. Thus,
arg k iT N1 π. (7) It remains to estimate N. To this end, we define
1
.2
k k k
g z ziT ziT
Then
1
Re
,2
k k k k
g iT iT iT
for
b c,
0k iT
so that if , then
0g .
Now let R2 c b and R> max
r1, 2
that T > 2RR , and
choose T large enough so . Then
1
> >
ziT R R for zc <R, showing that no zeros or poles of k
ziT
are located in zc <R. Thus, both k
ziT
and gk
z are analytic in<
zc R. Letting nˆc k,
r denote the number of zeros of gk
z in z c r, we have
2 , 2 ,
0
2
, ,
ˆ ˆ
ˆ ˆ log 2
d d
. d
R c k R c k R R
c k R c k
n r n r
r r
r r
r
n R n R
r
By Jensen’s formula
2 , 2π
0 0
ˆ 1
log 2 e d lo 2π
d g
R c k i
k k
n r
r g c R g
r
c ,and so
2π
, 0
log 1
ˆ log 2 e .
2πlog 2 d log2
k i
c k k
g c
n R
g c R (8)
1
1
2 1
2,1
1 1 1 1 1
1
1 1 1 1 1 1
1 Re
1 1 1 1 ,
d
k k
T k
c T k it
k n n k
k k
k
c c
k n n k n
k
a n a n t
I
k n n P c it n n
a n a n
O
k n n k n
By (5), log gk
coperty of L
is bounded. Further, it is clear
from a pr functions that we have
B,as ,as ;L s A t t t
for some positive absolute numbers A B,
estimate
in any vertical
strip of bounded width. The same must hold for
k
g z as well. Thus, the integral in (8) is O
logT
,implying th O
logT
. Since the interval
at n
ˆc k,
R
2
, , ,
b c D c R D c R , it follows that
ˆ
c k, log
.N n R O T
With this bound, we integrate (7) to deduce that
3, arg d 1 πd log .
c c
k
b b
1, , 4
j
k
I
it
N O TAs previously noted, we may bound I4,k in the same
way. Thus, we attain the desired bounds for and k1, 2. Consequently, the first part of t
is proved by using (4).
Proof of (II). As in the pro
theorem, we conclude that there exists a real number he theorem
of of the first part of the
0
for which the real parts R of all R-values satisfy
0
<
R
; and also, there exist B T, > 0 for each rational function R such that no zeros of
s 0L s R lie in the quarter-plane <B t, >T. As before, we define the rectangle
s it b: c T, t 2T
where b c T, ,
are parameters satisfying
0
r
< 1, > max 1, , > max , 1
b B c b T 1 T .
ding as in the proof of the first part of the theorem, we see that
Procee
2
1 ,
2
2π log
arg d arg 2 d
:
T
R T T k
c c
k k
b
j k j
i b i b it
iT O T
I I T
for where is defined as in (1). In the
equatio e, we n te that we have chosen to compute
4
O
b
2
dt
Tlog c it dt2
R
T T
iT
1, 2
k
n abov k
o
1
I separately. Indeed, this is the only estimate that we the integrals
will need. For Ij k,
pr
, and
, the bounds given as in the oo t part
eorem still hold. First, integral
unchanged. On the other hand, the integrals
have changed by our choice of but, as we ha
as before, we still have the desired bound
requirement is that we consid in a verti
of fixed width, which we have case.
We now bound
2,3, 4
j f of the firs
2,k is
1, 2
k h
of the t I
3,k, 4,k
I I
ve done since the only cal strip b
er in this
,
L
1
I . Since , we have by the
functional equation in the defi function,
<b B nition of L
1
L
L s R s s L s R s 1 1
1
1 1
L .
L
L
R s
s L s
s L s
R
s L s
L
s
s
Taking logarithms, we get
log
log L log 1 log 1 .
L s R s
R s
s L s
L s
(9)
niformly in
Since, for t> 1, we have, u ,
2
12
log
π 1
exp 1 ,
4
L
L
it L
d
L
s
i d
Q t itd O
t
where , are two constants. It follows, for s it as t , that
1
2 2 2
2 2
π 1 1
log log exp 1 log
4 2
1 1 1
log log 1 log log .
2
L L
L
it L
d d
L
it d
L
i d
s Q t it O Q t
t
Q t O d t Q O
t t
We now consider the last term in (9). Since,
log 1
, limsup
log 2 L
t
L b it
b d t
and noting b< 0, we have for any > 0 and tT
12 b dLL b it t
for sufficiently large T. Then we see the quotient
1
2
t
1
L
b d
R b it R b it
O
L b it t
when b is large enough so that
1
deg 1.
2 L
R b d
Therefore, we find that
1log 1 R s O .
L s t
Integrating in light of these estimates, we see
log
2
2 2
log
log log
2
log 1 .
d
d
d
T T
T T T
L b it R b it t
t Q t
L b it t O T
first i
2
1 T
L
b d
The ntegral is log4 log
2L
T
d T T Q
e , and the
second integral is O
1ethod
for sufficiently large and
negative b by the m used to derive (6). Hence,
2
1
1 4
log log log .
2 L
T
I b d T T Q O T
e
With the estimates for the Ij k, ’s, we have proved the
se .
[1] J. Steuding,“Value Distribution of L-Functions,” Number
1877 in Lecture Notes in Mathematics, Springer, 2007.
[2] H. S. A. Potter, “The Mean Values of Certain Dirichlet
on
8.
plms/s2-46.1.467
cond part of the theorem
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Series I,” Proceedings Lond Mathematical Society, Vol.
46, No. 2, 1940, pp. 467-46 http://dx.doi.org/10.1112/
[3] E. C. Titchma nctions,” 2nd Edi-
