Rudnick and Soundararajan’s Theorem for Function Fields
In this paper we prove a function field version of a theorem by Rudnick and Soundararajan about lower bounds for moments of quadratic Dirichlet –functions. We establish lower bounds for the moments of quadratic Dirichlet –functions associated to hyperelliptic curves of genus over a fixed finite field in the large genus limit.
Key words and phrases:function fields and finite fields and hyperelliptic curves and lower bounds for moments and moments of –functions and quadratic Dirichlet –functions and random matrix theory
2010 Mathematics Subject Classification:Primary 11M38; Secondary 11G20, 11M50, 14G10
It is a fundamental problem in analytic number theory to estimate moments of central values of –functions in families. For example, in the case of the Riemann zeta function the question is to establish asymptotic formulae for
where is a positive integer and .
A believed folklore conjecture asserts that, as , there is a positive constant such that
Due to the work of Conrey and Ghosh  the conjecture above assumes a more explicit form, namely
is an integer when is an integer and is the number of ways to represent as a product of factors.
Asymptotics for are only known for , due to Hardy and Littlewood 
and for , due to Ingham 
Unfortunately the recent technology does not allow us to obtain asymptotics for higher moments of the Riemann zeta function. The same statement applies for the higher moments of other –functions. However, due to the precursor work of Keating and Snaith [14, 15] and, subsequently, due to the work of Conrey, Farmer, Keating, Rubinstein and Snaith , and Diaconu, Goldfeld and Hoffstein , there are now very elegant conjectures for moments of –functions.
The work of Katz and Sarnak [12, 13] associates a symmetry group for each family of –function and the moments are sensitive and take different forms for each one of these groups. In other words the conjectured asymptotic formulas for the moments of families of –function depends whether the symmetry group attached to the family is unitary, orthogonal or symplectic. For a recent and detailed discussion about a working definition of a family of –functions see .
We will typify the conjectures above by considering different families of –functions. For example, the family of all Dirichlet –functions , as varies over primitive characters , is an example of a unitary family, and it is conjectured that
where and is a positive constant. For a symplectic family of –functions we consider the quadratic Dirichlet –functions associated to the quadratic character , as varies over fundamental discriminants. In this case it is conjectured that
where and is a positive constant. And finally we consider the family of –functions associated to Hecke eigencuspforms of weight for the full modular group as varies in the set of Hecke eigencuspforms. This is an example of an orthogonal family and it is conjectured that
where is a positive constant, and
where denotes the Petersson inner product. For more details on Hecke eigencuspforms –functions see Iwaniec .
for positive integers . Titchmarsh [24, Theorem 7.19] had proved a smooth version of these lower bound for positive integer . The work of Heath–Brown  extends (1.12) for all positive rational numbers . Recently Radziwiłł and Soundararajan  proved that
for any real number and all large . For other families of –functions, as those given above, the lower bounds for moments were proved by Rudnick and Soundarajan in [19, 20] where they have established that
for a fixed natural number and all large primes . They also proved in  that
for any given natural number , and weight with . And for the symplectic family they showed that for every even natural number
where the sum is taken over fundamental discriminants . Radziwiłł and Soundararajan  pointed out that their method may easily be modified to provide lower bounds for moments to the case of –functions in families, for any real number .
2. Main Theorem
Before we enunciate the main theorem of this paper we need a few basic facts about rational function fields. We start by fixing a finite field of odd cardinality with a prime. And we denote by the polynomial ring over and by the rational function field over .
The zeta function associated to is defined by the following Dirichlet series
where for and for . Surprisingly the zeta function associated to is a much simpler object than the usual Riemann zeta function and can be showed that
Let be a square–free monic polynomial in of degree odd. Then we define the quadratic character attached to by making use of the quadratic residue symbol for by
In other words, if is monic irreducible we have
We attach to the character the quadratic Dirichlet –function defined by
If , where
then the –function associated to is indeed the numerator of the zeta function associated to the hyperelliptic curve and therefore is a polynomial in of degree given by
This –function satisfies a functional equation, namely
and the Riemann hypothesis for curves proved by Weil  tell us that all the zeros of have real part equals .
The main result of this paper is now presented:
For every even natural number we have,
To avoid any misunderstanding concerning the notation and conventions presented in this paper it is necessary a note about the notation used in the theorem above and in the rest of this note. On the formula above the right-hand side of the main lower bound appears while is the summation variable on the left-hand side of that same formula. This is done because the function is constant within and so we can always write
In the case the reader feel uncomfortable with the above notation he/she can always remember that .
For simplicity, we will restrict ourselves to the fundamental discriminants , monic and . But the calculations are analogous for the even case, i.e., .
For every even natural number and or we have,
where and the prime number theorem for polynomials [18, Theorem 2.2] says that .
3. Necessary Tools
In this section we present some auxiliary lemmas that will be used in the proof of the main theorem. We start with:
Lemma 3.1 (“Approximate” Functional Equation).
Let . Then can be represented as
The proof of this Lemma can be found in [1, Lemma 3.3]. ∎
The following lemma is the function field analogue of Pólya–Vinogradov inequality for character sums.
Lemma 3.2 (Pólya–Vinogradov inequality for ).
Let be a non–principal Dirichlet character modulo such that is odd. Then we have,
The proof of this Lemma can be found in [9, Proposition 2.1]. ∎
The next lemma is taken from Andrade-Keating [1, Proposition 5.2] and it is about counting the number of square–free polynomials coprime to a fixed monic polynomial.
Let be a fixed monic polynomial. Then for all we have that
In this section we prove Theorem 2.1.
Let be a given even number, and set . We define
An application of Triangle inequality followed by Hölder’s inequality gives us that,
From (4) we have
We have that
If is not a perfect square then
And if is a perfect square then
for any .
If , then
For (4.8) we write
If then is a character sum to a non–principal character modulo . So using Lemma 3.2 we have that
Further we can estimate trivially the non–principal character sum by
Thus, if , we obtain that
After some arithmetic manipulations with the –terms we get that
Since , the error term above is . So,
Writing we see that
where represents the number of ways to write the monic polynomial as a product of factors.
We need to obtain an estimate for
To obtain the desired estimate we consider the corresponding Dirichlet series
with . Writing the above as an Euler product
we can identify the poles of . Similar calculations carried out in the classical case by Soundararajan and Rudnick [20, page 9] and Selberg [22, Theorem 2], and for function fields by Andrade and Keating [2, Section 4.3] shows us that has a pole at of order . Therefore we can write
where the first product has a pole at of order and the second product above (4.1) is convergent for and holomorphic in with
Thus we can use Theorem 17.4 from  to obtain the desired estimate. But we sketch below how this can be done. A standard contour integration (Cauchy’s theorem)
where is the boundary of the disc for some and a small circle about oriented clockwise. There is only one pole in the integration region and it is located at as can be seen from (4.1). To find the residue there, we expand both and in Laurent series about , multiply the results together, and pick out the coefficient of . After this residue calculation we obtain that
for a positive constant explicitly given by
In the end we obtain that
Therefore we can conclude that
4.2. Estimating .
It remains to evaluate and for that we need an “approximate” functional equation for . Using Lemma 3.1 with we have that
In the last equality in equation (4.2) the sums over and are exactly the same, with the only difference being the size of the sums, i.e., and . We estimate only the sum in the last equality and the result being the same for the sum just replacing by .
If is not a square then an application of Lemma 4.1 gives us that
With our choice of , we have that for not a square