Value-distribution of quartic Hecke -functions
Set and suppose that is a square-free algebraic integer with . Let denote the Hecke -function associated with the quartic residue character modulo . For , we prove an asymptotic distribution function for the values of the logarithm of
as varies. Moreover, the characteristic function of is expressed explicitly as a product over the prime ideals of .
Mathematics Subject Classification (2010): 11M41, 11R42
Keywords: value-distribution, logarithm of -functions, quartic characters
Let be a non-square integer such that and be the Kronecker symbol. In the early 1950s, S. Chowla and P. Erdős studied the distribution of values of quadratic Dirichlet -functions . They proved in [chowla-erdos] that when , then
exists and the distribution function is continuous and strictly increasing satisfying , . This result was further strengthened by P. D. T. A. Elliott for in [elliott-0].
A systematic study of the value-distribution of the logarithm and the logarithmic derivative of -functions on the half-plane has been carried out by Y. Ihara and K. Matsumoto (see for example [I-M1] and [I-M]). Based on the approach in [I-M], M. Mourtada and V. K. Murty proved ([M-M, Theorem 2]), assuming the Generalized Riemann Hypothesis (GRH) for , that for any , there exists a probability density function such that
Here denotes the set of the fundamental discriminants in the interval
If is a fundamental discriminant, , with denoting the Dedekind zeta function of and the Riemann zeta function. A. Akbary and A. Hamieh studied an analogue case of the above result of Mourtada and Murty. Let and be the ring of integers of , where . Let denote the set of square-free elements such that and . Further define , where is the Dedekind zeta function of . Then Akbary and Hamieh [AH, Theorem 1.4] proved that, without assuming GRH, for either or , there exists a corresponding probability density function such that for every ,
We note that can be decomposed as a product of Hecke -functions. In fact, it is shown in the paragraph below [AH, (2)] that
where is the Hecke -function associated with the cubic residue symbol .
Motivated by the above result, we consider the value-distribution of the logarithm of the product of quartic Hecke -functions in this paper. Set and , the ring of integers of . Let
In the same spirit as (1.1), we define
where is the Hecke -function associated with the quartic residue symbol . Our result is
Then there is a smooth density function such that
Furthermore, can be constructed as the inverse Fourier transform of the characteristic function
Our proof of Theorem 1.1 closely follows the treatment of Theorem 1.4 in [AH]. We shall therefore skip some of the details in the proof when the arguments are similar. Using arguments analogues to those in [AH, Section 2], we see that Theorem 1.1 is the consequence of the following two propositions which are quoted from [AH].
Fix and . We have
where is defined in (1.2) and henceforth indicates that the sum is over for which .
Let be given and be fixed. For sufficiently large values of , we have
where is a positive constant and can be chosen depending on values of and .
The following notations and conventions are used throughout the paper.
or means for some unspecified positive constant .
denotes an arbitrary positive number, whereas denotes a fixed positive constant.
and denote the ring of integers of .
The Gothic letters , , represent ideals of .
The norm of an integer is written as . The norm of an ideal is written as .
denotes the Möbius function on .
is the Dedekind zeta function for the field .
2.1. Distribution and characteristic functions
A function is said to be a distribution function if is non-decreasing, right-continuous with and . For example,
where is non-negative and . In this case, called tthe density function of . The characteristic function of , , is the Fourier transform of the measure , i.e.
For a more detailed discussion, we refer the reader to [AH] and the references therein.
2.2. Quartic residue symbol
The symbol is the quartic
residue symbol in the ring . For a prime
with , the quartic character is defined for , by , with . When , we define
. Then the quartic character can be extended
to any composite with multiplicatively. We extend the definition of to by setting .
Note that in , every ideal co-prime to has a unique generator congruent to 1 modulo . Such a generator is called primary. Recall that [Lemmermeyer, Theorem 6.9] the quartic reciprocity law states that for two primary integers ,
From the supplement theorem to the quartic reciprocity law (see for example, Lemma 8.2.1 and Theorem 8.2.4 in [BEW]), we have for being primary,
It follows that for any ,
The above shows that is trivial on units, hence it can be regarded as a primitive quartic character of the -ray class group of when is square-free.
2.3. Evaluation of and
As , we have for ,
and denotes the residue of at , denotes the -ray class group of .
We deduce from Lemma 2.4, by setting , that as ,
2.5. A zero density theorem
For , the Hecke -function associated with is defined by the Dirichlet series
can be analytically continued to the entirety of and satisfies a functional equation relating its values at and at . We shall need the following zero density theorem for .
[BGL, Corollary 1.6] For , and , let be the number of zeros of in the rectangle , . Then
2.7. The large sieve with quartic symbols and a Pólya-Vinogradov type inequality
In the course of the proof of Theorem 1.1, we need the following large sieve type inequality for quartic residue symbols, which is a special case of [BGL, Theorem 1.3] and an improvement of [G&Zhao, Theorem 1.1]:
For and be an arbitrary sequence of complex numbers, we have
where means that the summation runs over the square-free elements of .
We shall also need the following Pólya-Vinogradov type inequality for -ray class characters of .
[G&Zhao, Lemma 3.1] Let and be a non-trivial character (not necessarily primitive) of -ray class group of . Then for and , we have
2.10. A Dirichlet series representation for
For and any non-negative integer , we define the function by and for ,
This implies that
We further define the arithmetic function on the integral ideals of as follows:
Similar to [AH, Lemma 4.1], the following Lemma gives a Dirichlet series representation for .
Let and with . Then
where is given in (2.6). Moreover, the above series is absolutely convergent.
We omit the proof of Lemma 2.11 as it is similar to the proof of [AH, Lemma 4.1]. Moreover, It is shown in [I-M, p. 92] that for any and all , we have
3. Proof of Proposition 1.2
To establish Proposition 1.2, we first prove, in the next three section, Proposition 3.1 which gives a Dirichlet series (see (3.1)) representation for the limit in Proposition 1.2. Then in Section 3.7, we prove Proposition 3.8 which renders a product representation for the afore-mentioned Dirichlet series to show that it is the same as defined in (1.2).
Fix for some . Then for all we have
Here is given by the following absolutely convergent Dirichlet series
3.2. Application of the zero density estimate
Let be fixed and be the rectangle with the vertices and .
Let be the set consisting of such that does not vanish in . Also, set . Note that and vary with , , and .
Using arguments similar to those in the proof of [AH, Lemma 4.3], we see that for as fixed in Proposition 3.1 and a sufficiently small , we have
Furthermore, for , , and , we use Lemma 2.11 and arguments similar to those in the proof of [AH, Lemma 4.4] to derive the following lemma giving a representation of as a sum of an infinite sum and a certain contour integral.
Let be given with . Suppose that . If , then
where is the contour that connects, by straight line segments, the points , , , , and .
Inserting the above lemma into (3.2), we get, for ,
Letting and for , using arguments analogues to those in [AH], we deduce that
3.4. Evaluation of (I)
It still remains to prove an asymptotic formula for given in (3.4).
with defined in (3.1). Then
for any sufficiently small .
Starting with (3.4),
Rearranging the above sum over and with and recalling that if is prime to , we obtain
The part of (3.7) contributed by the fourth powers is
with an error that is . Here is defined in Lemma 2.4. Choose small enough so that . Inserting the formula
into (3.8) and shifting the line of integration to , we conclude the contribution of fourth powers to is
To estimate the contribution of non-fourth powers to , we write and with being primary for ideals and with . We then have, by the quartic reciprocity law (2.1),
We split the last sum above into two parts to get
Here is a parameter to be optimized later, (respectively ) is the multiplicative inverse of (respectively ) modulo and denotes summation over square-free elements of .