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 [
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 [
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 [
in the rectangle 
where 
are parameters satisfying
. Thus,
where the given logarithm is defined as in Littlewood’s argument principle [
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
To connect this integral with Littlewood’s argument principle [
guarantees that
In light of (2) and because the quantity given in (3) is imaginary-valued, we get for 
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 [
,
Hence, 
uniformly in 
.
We next move to estimate
. For sufficiently large positive real number
, we have
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,
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
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
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.