Value Distribution of L-Functions with Rational Moving Targets ()
1. Introduction
We define the class
to be the collection of functions
satisfying Ramanujan hypothesisAnalytic continuation and Functional equation. We also denote the degree of a function
by
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.
Theorem. Assume that
and
is a rational function with
. Let the roots of the equation
be denoted by
. Then
(I) For any
,

(II) For sufficiently large negative
,

Proof of (I). It is known that if
, then

where
is the index of the first non-zero term of the sequence of
,
with
. Since
, there exists
such that
for
. It follows that
for all real part of zeros of the function
. We set
where the degrees of
are
, respectively; and define

Thus, there is
such that
is analytic in the region
since
is a meromorphic function in
with the only pole at
. We apply Littlewood’s argument principle [3] to
in the rectangle
where
are parameters satisfying
. Thus,

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)
Without loss of generality, we may assume that
whenever
since we can always write
for
due to our choice of the parameters which define the rectangle
. However, the modification will guarantee in the case of
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

where the
terms are the integrals of the maximum contribution from writing
as a sum of logarithms. By our choice of
, both
and
are analytic in 
Hence, Cauchy’s Theorem gives
(2)
To connect this integral with Littlewood’s argument principle [3], we note that the definition of
guarantees that
(3)
In light of (2) and because the quantity given in (3) is imaginary-valued, we get for 
(4)
for instance.
We now estimate
. For
large enough, we have for
(since
),

Then for
large enough,
, we find in a similar fashion that

Since we have the same estimate for
, we find that

where the final bound follows from Jensen’s inequality. It is known [2] that for
,

Hence,
uniformly in
.
We next move to estimate
. For sufficiently large positive real number
, we have
(5)
so

since
. Furthermore,

Since we may take
large enough so that
, we may write
using a Taylor series expansion in the rectangle
. For
, we have after taking real parts that

We now observe that for sufficiently large T and some constant M we have

for
and

for sufficiently large
. In light of these bounds and the definition of
, we have (6)
where the last equality holds because
could be sufficiently large. Replacing
by
in the above computations, we see analogously that
.
Finally, we estimate
and
. We show the computation for
explicitly and note that the bound for
follows analogously. We first suppose that
has exactly
zeros for
. Then, there are at most
subintervals, counting for multiplicities, in which
is of constant sign. Thus,
(7)
It remains to estimate
. To this end, we define

Then

so that if
for
, then
.
Now let
and
, and choose
large enough so that
. Then
for
, showing that no zeros or poles of
are located in
. Thus, both
and
are analytic in
. Letting
denote the number of zeros of
in
, we have

By Jensen’s formula

and so
(8)
(6) |
By (5),
is bounded. Further, it is clear from a property of
functions that we have

for some positive absolute numbers
in any vertical strip of bounded width. The same estimate must hold for
as well. Thus, the integral in (8) is
, implying that
. Since the interval
, it follows that

With this bound, we integrate (7) to deduce that

As previously noted, we may bound
in the same way. Thus, we attain the desired bounds for
and
. Consequently, the first part of the theorem is proved by using (4).
Proof of (II). As in the proof of the first part of the theorem, we conclude that there exists a real number
for which the real parts
of all
-values satisfy
; and also, there exist
for each rational function
such that no zeros of
lie in the quarter-plane
. As before, we define the rectangle
where
are parameters satisfying
.
Proceeding as in the proof of the first part of the theorem, we see that

for
where
is defined as in (1). In the equation above, we note that we have chosen to compute
separately. Indeed, this is the only estimate that we will need. For the integrals
,
and
, the bounds given as in the proof of the first part of the theorem still hold. First, integral
is unchanged. On the other hand, the integrals
have changed by our choice of
, but, as we have done as before, we still have the desired bound since the only requirement is that we consider 
in a vertical strip of fixed width, which we have in this case.
We now bound
. Since
, we have by the functional equation in the definition of
function,

Taking logarithms, we get
(9)
Since, for
, we have, uniformly in
,

where
are two constants. It follows, for
as
, that

We now consider the last term in (9). Since,

and noting
, we have for any
and 

for sufficiently large
. Then we see the quotient

when
is large enough so that

Therefore, we find that

Integrating in light of these estimates, we see

The first integral is
, and the second integral is
for sufficiently large and negative
by the method used to derive (6). Hence,

With the estimates for the
’s, we have proved the second part of the theorem.
NOTES