Square-full polynomials in short intervals and in arithmetic progressions

We study the variance of sums of the indicator function of square-full polynomials in both arithmetic progressions and short intervals. Our work is in the context of the ring $F_{q}[T]$ of polynomials over a finite field $F_{q}$ of $q$ elements, in the limit $q\rightarrow\infty$. We use a recent equidistribution result due to N. Katz to express these variances in terms of triple matrix integrals over the unitary group, and evaluate them.


Background
A positive integer n is called a square-full number if p 2 |n for every prime factor p of n. Denote by α 2 the indicator function of square-full numbers, i.e. Let A(x) be the number of square full integers not exceeding x. In 1935, Erdös and Szekeres [5] proved Bateman and Grosswald [1] improved this result in 1958. They obtained where c is a positive absolute constant. They also made the observation that any improvement of the exponent 1 6 would imply that ζ (s) = 0 for s > 1 − δ (δ > 0). There is a very long history of studies and conditional improvements (assuming RH) of the error term in the above formula (see [2][3][4]15,[19][20][21][22] Various authors used exponential sum techniques to reduce the lower bound on H for which this asymptotic is valid. Heath-Brown [7] proved it with for H > x η+ with η = 0.6318 . . . Liu [16] proved it with η = 0.6308 . . . Filaseta and Trifonov [6] found a simpler approach, using real instead of complex analysis, and obtained the exponent η = 0.6282 . . . In [8], Huxley and Trifonov improve this to H ≥ 1 x 5/8 (log x) 5/16 . Concerning the distribution of square full numbers in arithmetic progressions, the most recent result is due to Munsch [17]. By evaluating character sums, he showed that n≤x n=a mod q with X 2 and X 3 being the set of all quadratic and cubic characters mod q respectively and L(s, χ) is the L-function attached χ. The goal of this paper is to study the fluctuations of the analogous sums in the function field settings. Namely, we study the variance of the sum of α 2 in arithmetic progressions and in short intervals, in the context of the ring F q [T ] of polynomials over a finite field F q of q elements, in the limit q → ∞. In our setting we succeed in giving definitive results in both cases.
Our approach involves converting the problem to one about the correlation of zeros of a certain family of L-functions, and then using an equidistribution result of Katz which holds in the limit q → ∞.
2 Square-full polynomials Let F q be a finite field of an odd cardinality q, and let M n be the set of all monic polynomials of degree n with coefficients in F q . In analogy to numbers, we say that f ∈ M n is a squarefull polynomial if for every polynomial P ∈ M n that divides f , P 2 also divides f . We denote by α 2 the indicator function of square-full polynomials, i.e. 0 1 7 ) 3 : 3 Page 3 of 18 The mean value of α 2 over all monic polynomials is defined to be The generating function for the number of monic square-full polynomials of degree n, i.e.
where Z(u) is the zeta function of F q [T ] (also set ζ q := Z(q −s )) , given by the following product over prime polynomials in F q [T ] By expanding we have Therefore, for n ≥ 6 the coefficient of u n is given by (2.7) Therefore in the limit of q → ∞ we get (2.8)

Arithmetic progressions
Let Q ∈ F q [T ] be a squarefree polynomial of a positive degree. The sum of α 2 over all monic polynomials of degree n lying in the arithmetic progressions f = A mod Q is (2.9) The average of this sum S α 2 ;n;Q (A) when we vary A over residue classes coprime to Q is In Sect. 4 we will consider the variance of S α 2 ;n;Q which is defined to be the average of the squared difference between S α 2 ;n;Q and its mean value |S α 2 ;n;Q − S α 2 ;n;Q | 2 .
(2.11) proving the following theorem: Theorem 2.1 Let Q be a prime polynomial of degree bigger than 3, and set N := deg Q−1, then in the limit q → ∞ the following holds: Var(S α 2 ;n;Q ) ∼ q n 2 (Q) , for 2N + 1 < n even Note that the restriction that Q is a prime is for simplicity only and we can also do the more general case of squarefree Q in the same way.

Short intervals
A "short interval" in F q [x] is a set of the form where A ∈ M n and 0 ≤ h ≤ n − 2. The norm is The cardinality of such a short interval is #I(A; h) = q h+1 =: H . (2.14) To facilitate comparison between statements for number field results and for function fields, we use a rough dictionary:  The conditions one needs to place on n in both Theorems 2.1 and 2.2 are not obvious to begin with. They will follow eventually because we express the variance in both cases in terms of zeros of L-functions, which are known to be polynomials in the function field settings.

Dirichlet characters
Let Q(T ) ∈ F q [T ] be a polynomial of positive degree. A Dirichlet character modulo Q is a homomorphism One can extend χ to F q [T ] by defining it to vanish on polynomials which are not coprime to Q. We denote by (Q) the group of all Dirichlet characters modulo Q. Note that | (Q)| is the Euler totient function (Q). A Dirichlet character needs then to satisfy the following: The orthogonality relations for Dirichlet characters are: Page 6 of 18 A character χ is primitive if there is no proper divisor Q |Q such that χ(f ) = 1 whenever f is co-prime to Q and f = 1 mod Q . A character χ is called "even" if it acts trivially on the elements of F q , i.e. if χ(cf ) = χ(f ) for all 0 = c ∈ F q . Therefore, the number of even characters is given by ev (Q) = (Q) q−1 . For example there are q m−1 even character mod T m . Out of this there are O(q m−2 ) nonprimitive even characters mod T m (see subsection 3.3 in [13]) . A character is called "odd" if it is not even.
Define the following: Next, we will check the proportion of the set d−prim (Q) in the group of all characters mod Q , (Q).
be a square-free polynomial, then in the limit of a large field size q → ∞, Proof By subsection (3.3) in [13] we have for some constant c. Note that a character χ does not lie in d-prim (Q) if χ d does not lie in prim (Q). In that case, χ d lies in (Q)/ prim (Q). Now consider the map χ → χ d . Its kernel is of cardinality #{χ|χ d = 1}. Since every χ is a product of characters χ j of the following bound holds: Since every character χ such that χ d is not primitive can be written as a product of a non primitive character and an element of the kernel, we get that Note that equation (3.25) in [13] asserts that as q → ∞, almost all characters are primitive and odd in the sense that odd prim (Q) Hence, exactly as before, we may also show that 0 1 7 ) 3 : 3 Page 7 of 18 be a square-free polynomial, then in the limit of a large field size q → ∞, Next, we will prove a short lemma stating that under certain restrictions on the characteristic of the field, the primitivity of χ and χ d is equivalent when χ is an even character mod T m . This lemma will be useful later on in Sect. 5.

Lemma 3.3 Let d be an integer co-prime to (Q).
Then the map χ → χ d is an automorphism of the group of characters mod Q, i.e. an automorphism of (Q).
Proof The map is clearly an homomorphism since the group is abelian. Now, d is co-prime to the order of the group therefore there aren't any elements whose order dividing d and hence the kernel of the map is trivial.

Lemma 3.4 Let χ be an even Dirichlet character mod T m , and let d be an integer s.t. d < p when p is the characteristic of the field F q . Then χ is a primitive character if and only if χ d is a primitive character.
Proof The order of the subgroup of even characters mod T m is ev (T m ) = q m−1 . Taking d < p when p is the characteristic of the field F q gives d co-prime to (T m ), in which case the above lemma applies.

Dirichlet L-functions
Here we review some standard background concerning Dirichlet L-functions for the rational function field; see, for example [18], subsection 3.4 in [13], or section 6 in [14].
The L-function associated to a Dirichlet character χ mod Q is defined as the following product over prime polynomials P ∈ F q [T ] The product is absolutely convergent for |u| < 1/q. For χ = χ 0 the trivial character mod Q is a polynomial of degree deg Q ≥ 2 and χ is a nontrivial character mod Q, then the L-function associated to χ i.e. L(u, χ) is a polynomial in u of degree at most deg Q − 1. If χ is even then L(u, χ) has a trivial zero at u = 1. Now, we may factor L(u, χ) in terms of the inverse roots for which the Riemann hypothesis, proved by Weil, asserts that for each (nonzero) inverse root, either |α j (χ)| = 1 or for the trivial zero at 1. Thus, we may write α j (χ) = q 1/2 e i j , and the L-function (for a primitive character χ) is . (3.13) where N = deg Q − 1 and λ χ = 0 for odd character χ. For even χ we have N = deg Q − 2 and λ χ = 1. The unitary matrix χ ∈ U (N ) determines a unique conjugacy class which is called the unitarized Frobenius matrix of χ.

Katz's equidistribution results
The main ingredients in our results on the variance are equidistribution and independence results for the Frobenii χ due to N. Katz. Theorem 3.8 [12] If in addition we restrict the characteristics of the fields F q is bigger than 6, then the set of conjugacy classes ( χ 2 , χ 3 , χ 6 ) with χ running over all characters such that χ 2 , χ 3 , χ 6 are primitive odd characters modulo Q, become equidistributed in the space of conjugacy classes of the product U (m − 1) × U (m − 1) × U (m − 1) as q goes to infinity.

The mean value
Given a polynomial Q ∈ F q [T ] the average of S α 2 ;n;Q (A) when we vary A over residue classes co-prime to Q (see (2.9)) equals to To evaluate this consider the generating function Page 9 of 18 By expanding and comparing coefficients it is clear that the leading order coefficient in q (we are interested in q → ∞) comes from Z(u 2 )Z(u 3 ) Z(u 6 ) . Therefore, by 2.8, we have S α 2 ;n;Q ∼ q n/2 (Q) .
In the rest of this section we will evaluate the variance of S α 2 ;n;Q i.e. the average of the squared difference between S α 2 ;n;Q and its mean value.

The case of small n
See also subsection 4.2 in [14]. If n < deg Q then there is at most one polynomial f ∈ F q [T ] of degree n such that f = A mod Q. In this case, when q → ∞ Var(S α 2 ;n;Q ) ∼ q n (Q) α 2 n .

The quadratic character and the cubic character
Next, we evaluate M(n; α 2 χ) for χ = χ 2 a quadratic character and for χ = χ 3 a cubic character. We assume here for simplicity that Q is prime polynomial. The sum M(n; α 2 χ) given by (4.8), is the coefficient of u n in the expansion of L(u 2 ,χ 2 )L(u 3 ,χ 3 ) . Thus for χ = χ 2 , the generating function of M(n; α 2 χ 2 ) has the following form: (4.12) and that therefore by expanding and comparing coefficients while bearing in mind the Riemann hypothesis (3.12), we can see that for an even n the leading order coefficient in q come from Z(u 2 ) and for odd n it comes from Z(u 2 ) and L(u 3 , χ 2 ). Thus we get α j (χ 2 ) n odd, (4.14) where α j (χ 2 ) are the inverse roots of L(u, χ 2 ). For χ = χ 3 , the generating function of M(n; α 2 χ 3 ) has the following form: as before we get where α j (χ 3 ) are the inverse roots of L(u, χ 3 ). For χ = χ 0 , χ 2 , χ 3 mod Q, the following bound holds:

Average of the sum
Proof The sum M(n; α 2 χ) given by (4.8), is the coefficient of u n in the expansion of . For an odd primitive characters χ mod Q, we use the Riemann Hypothesis (3.12) (Weil's theorem) to write Since the coefficients of the characteristic polynomial of an N ×N matrix with eigenvalues λ 1 , . . . , λ n are the elementary symmetric functions 1≤i 1 <...<i r λ i 1 · · · λ i r which give the character of the exterior power representation, we may write (4.20) and in the same way Abbreviate as follows: j (χ 2 ) l (χ 3 )Sym k (χ 6 ).
Next, we want to evaluate 2j+3l+6k=n  Proof In order to find the leading order term of S(n) we need first to take the maximal possible j and then the maximal possible l which satisfy 2j + 3l + 6k = n, 0 ≤ j ≤ N, 0 ≤ l ≤ N (note that k will then be determined).
In the first case 0 ≤ n ≤ 2N if 2 divides n then j = n/2, l = 0, k = 0 will clearly give the leading order term. If 2 does not divide n then j = (n − 3)/2, l = 1, k = 0 will give the leading order term.
In the second case 2N < n ≤ 5N , we write j = N − i j and then we have n − 2N = 6k +3l −2i j and so clearly the values for k, l, i j that will give the leading order term, depend on the value of n − 2N mod 3 (or equivalently n + N mod 3). Here our first priority is to minimize i j and then to maximize l. The leading order term will be given by q . In the last case 5N < n, we write j = N − i j and l = N − i l , then we have n − 5N = 6k − 3i l − 2i j and so clearly the values for k, i l , i j that will give the leading order term, depend on the value of n − 5N mod 6. Here our first priority is to minimize i j and then to minimize i l . The leading order term will be given by q n+7N −3i l −4i j 6 . Note that in the notations of Eqs. (4.26) and (4.27) we have 4i j + 3i l = λ n

Proof of Theorem 2.1
Recall that by Lemma 3.2, we have that the number of characters which are not in q ) when q → ∞. Therefore, by using the formula for the variance that was given in (4.10) we may write Var(S α 2 ;n;Q ) = 1 (Q) 2 where λ 3 = 2 if |Q| = 1 mod 3, and zero otherwise. Note that we assume Q is prime and that the characteristic of the field F q is odd therefore there is one quadratic character mod Q and either two or zero cubic characters, depending on whether |Q| = 1 mod 3 or not. 0 1 7 ) 3 : 3 Page 13 of 18 For the first sum in (4.28), use (4.17) to have (4.29) We can use now the equidistribution result given in Theorem 3.8, to have tr Sym k 1 (U 3 )tr Sym k 2 (U 3 ) dU 3 (4.30) It is well known that j are distinct irreducible representations of the unitary group U (N ), and hence one gets The contribution from the second and third summands was evaluated in Subsect. 4.4. Adding up everything and checking for the leading order terms by using Lemma 4.2 finishes the proof.

The mean value
The mean value of N α 2 ;h when we average over A ∈ M n is where H = #I(A; h) = q h+1 . In the rest of this section we will evaluate the variance of N α 2 ;h i.e. the average of the squared difference between N α 2 ;h and its mean value.

An expression for the variance
To begin the proof of Theorem 2.2, we express the variance of the short interval sums N α 2 ;h in terms of sums of the function α 2 , twisted by primitive even Dirichlet characters, similarly to what was done in the previous section. To compute the variance, we need to obtain an expression for M(n; α 2 χ). Consider the generating function ∞ n=0 M(n; α 2 χ)u n = L(u 2 , χ 2 )L(u 3 , χ 3 ) L(u 6 , χ 6 ) . (5.7) For an even primitive characters χ mod T n−h , L(u, χ) has a trivial zero at u = 1, hence we may write ) for χ mod T n−h such that χ 2 , χ 3 , χ 6 are primitive and even can be written as follows: j (χ 2 ) l (χ 3 )Sym k (χ 6 ). Now back to the variance formula (see (5.3)), we can split the sum into two parts: the sum over χ = χ 0 mod T n−h , χ ∈ ev prim (T n−h ) and the sum over even non-primitive characters mod T n−h . We start by considering the first sum which will give the main term since most of the even characters are also primitive. With the second sum which will give an error term we deal later. Note: by Lemma 3.4 it is enough to split to these sums, and we can still use (5.9) and (5.8) for χ 2 , χ 3 , χ 6 .
For χ even and primitive, consider the inner sum in the variance formula: