Abstract
Let Q be a positive defined matrix and . The Epstein zeta-function , , is defined, for , by the series and is meromorphically continued on the whole complex plane. Suppose that is even and is a differentiable function with a monotonic derivative. In the paper, it is proved that , converges weakly to an explicitly given probability measure on as .
MSC:
11M46; 11M06
1. Introduction
It is well known that the Riemann zeta-function
shows analytic continuation to the whole complex plane, except for a simple pole at the point , and satisfies functional equation
where denotes the Euler gamma-function. The majority of other zeta-functions also have similar equations, which are referred to as the Riemann type. Epstein in [] raised a question to find the most general zeta-function with a functional equation of the Riemann type and introduced the following zeta-function. Let Q be a positive defined quadratic matrix, and for . Epstein defined, for , the function
continued analytically it to the whole complex plane, except for a simple pole at the point with residue , and proved the functional equation
In [], Bohr and Jessen proved a probabilistic limit theorem for the function . We recall its modern version. Denote by the Borel -field of the topological space , and by measA the Lebesgue measure of a measurable set . Then, on , there exists a probability measure such that, for ,
converges weakly to as (see, for example, [] (Theorem 1.1, p. 149). In [], the latter limit theorem was generalized for the Epstein zeta-function with even and integers . Namely, on , there exists an explicitly given probability measure such that, for ,
converges weakly to as .
For the function , more general limit theorems are also considered. In place of (1), the weak convergence for
with certain measurable function is studied. For example, theorems of such a kind follow from limit theorems in the space of analytic functions proved in [].
Suppose that the function is defined for , is increasing to , and has a monotonic derivative satisfying the estimate
Denote the class of the above functions by .
The aim of this paper is to prove a limit theorem for
when . In place of one can consider
It is easily seen that the weak convergence of to as is equivalent to that of . Actually, if converges weakly to as , then
for every continuity set A of the measure . Since
we obtain that
i.e., converges weakly to as .
Now, suppose that (2) is true. Then
where as . Taking and summing the above equality over , we obtain, ue of -additivity of the Lebesgue measure,
Let . We fix such that
Then
Thus, taking and then , we find
This together with (3) shows that
i.e., converges weakly to as .
Since, in the case of the function occurs for large values of t, the study of sometimes is more convenient than that of . Therefore, we will prove a limit theorem for .
As in [], we use the decomposition []
where and are zeta-functions of certain Eisenstein series and of a certain cusp form, respectively. The latter decomposition and the results of [,]—see also []—imply that, for ,
where , , and are Dirichlet L-functions, and the series is absolutely convergent for . Equality (4) is the main relation for investigation of the function . Before the statement of a limit theorem, we construct a -valued random element connected to .
Let is the set of all prime numbers, , and
where for all . The infinite-dimensional torus is a compact topological Abelian group; therefore, the probability Haar measure can be defined on . This gives the probability space . Denote by the pth, , component of an element , and extend the function to the whole set by the formula
On the probability space , for , define the -valued random element by
where
and
Now, denote by the distribution of , i.e.,
Because , for . Therefore, the second Euler product for Dirichlet L-function is convergent for almost all and defines a random variable.
The main the result of the paper is the following theorem.
Theorem 1.
Suppose that , and is fixed. Then converges weakly to the measure as .
Since the representation (4) depends on Q, the random element depends on Q. Thus, the limit measure also depends on Q.
2. Some Estimates
We will consider the measure ; therefore, we suppose that with large T. Let be a Dirichlet character modulo q, and be a corresponding Dirichlet L-function.
Lemma 1.
Suppose that and is fixed. Then, for ,
Proof.
It is well known that, for fixed ,
An application of the mean value theorem, in view of (5), gives
where and is increasing. Thus, by the definition of the class ,
If is decreasing, then similarly we have
□
Let be a fixed number, and
where . Put
Then, by the exponential decreasing of , the latter series is absolutely convergent for with arbitrary finite . Define
Then the series for is absolutely convergent for . It turns out that approximates well in the mean the function . More precisely, we have the following result.
Lemma 2.
Suppose that and is fixed. Then
Proof.
Let be from the definition of ; denotes the Euler gamma-function, and
Then, the Mellin formula
leads, for , to
Denote by the principal Dirichlet character modulo q. Since the function is entire for , and has a simple pole at the point with residue
the residue theorem and (6) give
where and
Therefore,
Hence, we have that
where
and
It is well known that, uniformly in with arbitrary ,
Therefore,
and
Suppose that and is such that . Then, in view of (8) again,
Therefore, Lemma 1 implies
This, (9) and (7) show that, for fixed ,
Since , we have . Therefore, for ,
Hence,
□
3. Limit Theorems
We divide the proof of Theorem 1 into lemmas that are limit theorems in some spaces.
We start with a lemma on the torus . For , define
Lemma 3.
Suppose that . Then converges weakly to the Haar measure as .
Proof.
We will apply the Fourier transform method. Let , be the Fourier transform of , i.e.,
where “*” indicates that only a finite number of integers are distinct from zero. Thus, by the definition of ,
Obviously,
Now, suppose that . Since the set is linearly independent over the field of rational numbers, we have
Then
where . Since ,
as . Therefore,
Similarly, we find that
Thus, (10)–(12) show that
Since the right-hand side of the latter equality is the Fourier transform of the Haar measure , the lemma is proved. □
For , define
To prove the weak convergence for as , consider the function given by the formula
where
and
is the Dirichlet series for . Clearly, the above series are absolutely convergent for . The absolute convergence of the series for implies the continuity for the function . Therefore, the function is -measurable, and we can define the probability measure , where
Lemma 4.
Suppose that and is fixed. Then, converges weakly to as .
Proof.
By the definitions of , and , for all ,
Thus, . Therefore, the lemma is a consequence of Theorem 5.1 from [], continuity of and Lemma 3. □
The measure is very important for the proof of Theorem 1. Since is independent of the function , the following limit relation is true.
Lemma 5.
Suppose that is fixed. Then converges weakly to as .
Proof.
In the proof of Theorem 2 from [], it is obtained (relation (12)) that converges weakly to a certain measure , and, at the end of the proof, the measure is identified by showing that . □
For convenience, we recall Theorem 4.2 of []. Denote by the convergence in distribution.
Lemma 6.
Suppose that the space is separable, and -valued random elements , are defined on the same probability space with measure P. Let, for every k,
and
If, for every ,
then X.
Proof of Theorem 1.
Suppose that is a random variable defined on a certain probability space and distributed uniformly in . Since the function is continuous, it is thus measurable, and is a random variable as well. Denote by the complex valued random element having the distribution , and, on the probability space , define the random element
Then, in view of Lemma 4,
and, by Lemma 5,
Define one more complex-valued random element
Then, an application of Lemma 2 gives, for ,
This, relations (13) and (14) show that all hypotheses of Lemma 6 are satisfied. Therefore,
and this is equivalent to the assertion of the theorem. □
4. Concluding Remarks
By Bohr and Jessen’s works, it is known that the asymptotic behavior of the Dirichlet series can be characterized by probabilistic limit theorems. It turns out that Bohr–Jessen’s ideas can also be applied for the Epstein zeta-function whose definition involves a positive defined matric Q. We prove that, for any fixed ,
converges weakly to an explicitly given probability measure on as . Here is an increasing differentiable function such that
For example, can be a polynomials or the Gram function. We recall that the Gram function is the solution of the equation
where is the increment of the argument of the function along the segment connecting the points and . It is known [] that
and
as .
Author Contributions
Investigation, A.L. and R.M.; writing—original draft preparation, A.L. and R.M.; writing—review and editing, A.L. and R.M. All authors have read and agreed to the published version of the manuscript.
Funding
Renata Macaitienė is funded by the Research Council of Lithuania (LMTLT), agreement No. S-MIP-22-81.
Conflicts of Interest
The authors declare no conflict of interest.
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