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Squarefull polynomials in short intervals and in arithmetic progressions
 E. RodittyGershon^{1}Email author
 Received: 12 May 2016
 Accepted: 23 November 2016
 Published: 17 January 2017
Abstract
We study the variance of sums of the indicator function of squarefull polynomials in both arithmetic progressions and short intervals. Our work is in the context of the ring \(\mathbb {F}_q[T]\) of polynomials over a finite field \(\mathbb {F}_q\) 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.
1 Background
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 \(\alpha _{2}\) in arithmetic progressions and in short intervals, in the context of the ring \(\mathbb {F}_q[T]\) of polynomials over a finite field \(\mathbb {F}_q\) of q elements, in the limit \(q\rightarrow \infty \). 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 Lfunctions, and then using an equidistribution result of Katz which holds in the limit \(q \rightarrow \infty \).
2 Squarefull polynomials
2.1 Arithmetic progressions
Theorem 2.1
Let Q be a prime polynomial of degree bigger than 3, and set \(N:=\deg Q  1\), then in the limit \(q\rightarrow \infty \) the following holds:
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.
2.2 Short intervals
Theorem 2.2
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 Lfunctions, which are known to be polynomials in the function field settings.
3 Dirichlet characters and Katz’s equidistribution results
3.1 Dirichlet characters
 \(\Gamma _{prim}(Q)\) :

the set of all primitive characters mod Q, \(\Phi _{prim}(Q):=\Gamma _{prim}(Q).\)
 \(\Gamma _{prim}^{ev}(Q)\) :

the set of primitive even characters mod Q, \(\Phi _{prim}^{ev}(Q):=\Gamma _{prim}^{ev}(Q).\)
 \(\Gamma _{prim}^{odd}(Q)\) :

the set of primitive odd characters mod Q, \(\Phi _{prim}^{odd}(Q):=\Gamma _{prim}^{odd}(Q).\)
 \(\Gamma _{d\text {}prim}(Q)\) :

the set of all characters \(\chi \) mod Q such that \(\chi ^{d}\) (i.e. \(\chi \times \cdots \times \chi \)) is primitive for the fixed integer \(d>0\), \(\Phi _{d\text {}prim}(Q):=\Gamma _{d\text {}prim}(Q).\) Note that this is a subset of \(\Gamma _{prim}(Q).\)
 \(\Gamma _{d\text {}prim}^{d\text {}odd}(Q)\) :

the set of all characters \(\chi \) mod Q such that \(\chi ^{d}\) is primitive and odd for the fixed integer \(d>0\), \(\Phi _{d\text {}prim}^{d\text {}odd}(Q):=\Gamma _{d\text {}prim}^{d\text {}odd}(Q).\)
Next, we will check the proportion of the set \(\Gamma _{dprim}(Q)\) in the group of all characters mod Q ,\(\Gamma (Q).\)
Lemma 3.1
Proof
Now, if Q factors into k irreducible polynomials \(f_{i}\) of degree \(d_{i}\), \(\deg Q:=n=\sum _{i=1}^{k}d_{i}\) then \(\Phi (Q)=\prod _{i=1}^{k}(q^{d_{i}}1)\ge (q1)^{n}\). It follows that \(\Phi _{dprim}(Q) \ge (q1)^{n}d^{n}\cdot c\cdot q^{n1}\). Therefore, in the limit of \(q\rightarrow \infty \) we get (3.4). \(\square \)
Lemma 3.2
Next, we will prove a short lemma stating that under certain restrictions on the characteristic of the field, the primitivity of \(\chi \) and \(\chi ^{d}\) is equivalent when \(\chi \) 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 coprime to \(\Phi (Q)\). Then the map \(\chi \mapsto \chi ^{d}\) is an automorphism of the group of characters mod Q, i.e. an automorphism of \(\Gamma (Q)\).
Proof
The map is clearly an homomorphism since the group is abelian. Now, d is coprime 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. \(\square \)
Lemma 3.4
Let \(\chi \) 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 \(\mathbb {F}_q\). Then \(\chi \) is a primitive character if and only if \(\chi ^{d}\) is a primitive character.
Proof
The order of the subgroup of even characters mod \(T^{m}\) is \(\Phi ^{ev}(T^{m})=q^{m1}\). Taking \(d<p\) when p is the characteristic of the field \(\mathbb {F}_q\) gives d coprime to \(\Phi (T^{m})\), in which case the above lemma applies. \(\square \)
3.2 Dirichlet Lfunctions
Here we review some standard background concerning Dirichlet Lfunctions for the rational function field; see, for example [18], subsection 3.4 in [13], or section 6 in [14].
3.3 Katz’s equidistribution results
The main ingredients in our results on the variance are equidistribution and independence results for the Frobenii \(\Theta _{\chi }\) due to N. Katz.
Theorem 3.5
[9] Fix \(m \ge 4\). The unitarized Frobenii \(\Theta _{\chi }\) for the family of even primitive characters \(\mod T^{m+1}\) become equidistributed in the projective unitary group \(PU(m1)\) of size \(m1\), as q goes to infinity.
Theorem 3.6
[11] If \(m \ge 5\) and in addition the characteristics of the fields \(\mathbb {F}_q\) are bigger than 13, then the set of conjugacy classes \((\Theta _{\chi ^{2}}, \Theta _{\chi ^{3}}, \Theta _{\chi ^{6}} )\), \(\chi \) is even primitive character \(\mod T^{m+1}\), become equidistributed in the space of conjugacy classes of the product \(PU(m1) \times PU(m1)\times PU(m1)\) as q goes to infinity.
For odd characters, the corresponding equidistribution and independence results are
Theorem 3.7
[10] Fix \(m \ge 2\). Suppose we are given a sequence of finite fields \(\mathbb {F}_q\) and squarefree polynomials \(Q(T) \in \mathbb {F}_q[T]\) of degree m. As \(q \rightarrow \infty \), the conjugacy classes \(\Theta _{\chi }\) with \(\chi \) running over all primitive odd characters modulo Q, are uniformly distributed in the unitary group \(U(m1)\).
Theorem 3.8
[12] If in addition we restrict the characteristics of the fields \(\mathbb {F}_q\) is bigger than 6, then the set of conjugacy classes \((\Theta _{\chi ^{2}}, \Theta _{\chi ^{3}}, \Theta _{\chi ^{6}} )\) with \(\chi \) running over all characters such that \(\chi ^{2},\chi ^{3},\chi ^{6}\) are primitive odd characters modulo Q, become equidistributed in the space of conjugacy classes of the product \(U(m1) \times U(m1)\times U(m1)\) as q goes to infinity.
4 The variance in arithmetic progressions
4.1 The mean value
4.2 The case of small n
4.3 A formula for the variance
4.4 The quadratic character and the cubic character
4.5 Average of the sum \(\mathcal M(n;\alpha _{2}\chi )\)
Lemma 4.1
Proof
For \(\chi \ne \chi _{0},\chi _{2},\chi _{3}\) mod Q for which at least one of \(\chi ^{2},\chi ^{3},\chi ^{6}\) is not primitive or not odd, we still have \(L(u,\chi )=\prod _{j=1}^{\deg Q1}(1\alpha _{j}(\chi )u)\) with all the inverse roots \(\alpha _{j}(\chi )=q^{\frac{1}{2}}\) or \(\alpha _{j}(\chi )=1\) , and hence we obtain (4.18). \(\square \)
Lemma 4.2
Let \(N:=\deg Q  1\) then in the limit \(q\rightarrow \infty \),
Proof
In order to find the leading order term of \(\textit{S}(n)\) we need first to take the maximal possible j and then the maximal possible l which satisfy \(2j+3l+6k=n, 0\le j \le N, 0\le l \le N\) (note that k will then be determined).
In the first case \(0\le n \le 2N\) 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 \(j=(n3)/2,l=1,k=0\) will give the leading order term.
In the second case \(2N< n \le 5N\), we write \(j=Ni_{j}\) and then we have \(n2N=6k+3l2i_{j}\) and so clearly the values for \(k,l,i_{j}\) that will give the leading order term, depend on the value of \(n2N \mod 3\) (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 \(q^{\frac{n+Ni_{j}}{3}\frac{k}{2}}\) which gives \(q^{\left\lfloor \frac{n+N}{3}\right\rfloor }.\)
In the last case \(5N< n\), we write \(j=Ni_{j}\) and \(l=Ni_{l}\), then we have \(n5N=6k3i_{l}2i_{j}\) and so clearly the values for \(k,i_{l},i_{j}\) that will give the leading order term, depend on the value of \(n5N \mod 6\). Here our first priority is to minimize \(i_{j}\) and then to minimize \(i_{l}\). The leading order term will be given by \(q^{\frac{n+7N3i_{l}4i_{j}}{6}}\). Note that in the notations of Eqs. (4.26) and (4.27) we have \(4i_{j}+3i_{l}=\lambda _{n}\) \(\square \)
4.6 Proof of Theorem 2.1
5 The variance over short intervals
5.1 The mean value
5.2 An expression for the variance
To begin the proof of Theorem 2.2, we express the variance of the short interval sums \(\mathcal N_{\alpha _{2};h}\) in terms of sums of the function \(\alpha _{2}\), twisted by primitive even Dirichlet characters, similarly to what was done in the previous section.
Lemma 5.1
Proof
Now, in order to complete the proof, it remains to bound the contribution of the even characters which are nonprimitive to the variance. Note that we do not have quadratic or cubic characters here since \(\Phi ^{ev}(T^{m})=q^{m1}\) and we are considering the case of characteristic bigger than 13 (see Theorem 3.6), therefore there cannot be any even characters mod \(T^{m}\) of order 2 or 3. For even characters which are nonprimitive we still have the bound (4.18) with \(N=nh2\), and since their proportion is O(1 / q) in the set of even characters, then we can bound their contribution as in the previous section, therefore we skip the verification.
Acknowledgements
The author gratefully acknowledges support under EPSRC Programme Grant EP/K034383/1 LMF: LFunctions and Modular Forms. The author would like to thank Zeev Rudnick for suggesting this problem and to both Jon Keating and Zeev Rudnick for helpful discussions and remarks. The author would also like to thank Ofir Gorodetsky for an important observation, and to the referees for their comments.
Competing interests
The author declares that she has no competing interests.
Declarations
Open Access
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