Correlations of sums of two squares and other arithmetic functions in function fields

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by  Lior Bary-Soroker

CORRELATIONS OF SUMS OF TWO SQUARES AND OTHER
ARITHMETIC FUNCTIONS IN FUNCTION FIELDS
LIOR BARY-SOROKER AND ARNO FEHM
Abstract. We investigate a function field analogue of a recent conjecture on autocorre-
lationsofsumsoftwosquaresbyFreiberg,KurlbergandRosenzweig,whichgeneralizesan
older conjecture by Connors and Keating. In particular, we provide extensive numerical
7
1 evidence and prove it in the large finite field limit. Our method can also handle correla-
0 tions of other arithmetic functions and we give applications to (function field analogues
2 of)theaverageofsumsoftwosquaresonshiftedprimes,andtoautocorrelationsofhigher

n divisor functions twisted by a quadratic character.
a
J
5 1. Introduction
1
Westudyfunctionfieldanaloguesofconjecturesonautocorrelationsofsumsoftwosquares.

]
T
N 1.1. Correlations of arithmetic functions. Abasicstatisticalpropertyofanarithmetic
function ψ is its mean value; that is, the asymptotic as x → ∞ of
.
h
t 1 (cid:88)
a (1.1) (cid:104)ψ(n)(cid:105) = ψ(n).
m n≤x x
n≤x
[

More information is given by the cross-correlations of arithmetic functions ψ ,...,ψ ,
1 1 k
v which are defined at (h1,...,hk) ∈ Zk as the asymptotic as x → ∞ of
2
(cid:42) (cid:43)
9 (cid:89)k 1 (cid:88)
0 (1.2) ψ (n+h ) = ψ (n+h )···ψ (n+h ).
i i 1 1 k k
4 x
0 i=1 n≤x n≤x
.
1 In the special case when all the ψ are equal, we use the term autocorrelations. Some
i
0
of the most famous theorems and problems in number theory are about these statistical
7
properties for certain specific arithmetic functions, like the Hardy-Littlewood prime tuple
1
: conjecture, a quantitative version of the twin prime conjecture, which can be expressed in
v
i terms of the autocorrelations of the von Mangoldt function Λ.
X
r
a 1.2. Autocorrelation of sums of two squares. An integer n is a sum of two squares
if there exist x,y ∈ Z such that n = x2 + y2; i.e., it is a norm of the Gaussian integer
x+iy ∈ Z[i]. We let
(cid:40)
1, if there exist x,y ∈ Z such that n = x2 +y2,
(1.3) b(n) =
0, otherwise.
1
2 LIOR BARY-SOROKER AND ARNO FEHM
The study of the statistics of b(n) has a long history: Already Landau [Lan08] gives the
mean value of b as
1 (cid:88) 1
(1.4) (cid:104)b(n)(cid:105) = b(n) ∼ K · √ , x → ∞,
n≤x x logx
n≤x
where
1 (cid:89)
(1.5) K = √ (1−p−2)−1/2 ≈ 0.764
2
p≡3(mod4)
is the Landau-Ramanujan constant. The distribution of sums of two squares was then
studied intensively through the behavior of b in short intervals in many works by various
authors, including [Hoo74, Iwa76, Hoo94], see the introduction of [BBF17, §1.2] for a brief
history.
As for autocorrelations of b, one has lower and upper bounds of the right order of
magnitude for the pair autocorrelations:
1 1
(1.6) (cid:28) (cid:104)b(n)b(n+h)(cid:105) (cid:28) .
logx n≤x logx
For h = 1, the upper bound was proved by Rieger [Rie65] while the lower bound by
Indlekofer and Schwarz in [IS72, Ind74, Sch72] (see Kelly [Kel78] and Bantle [Ban86] for
short interval versions of the lower bound in (1.6)). Hooley [Hoo74] proved (1.6) for general
h (cid:54)= 0. As Hooley [Hoo71] asserts, determining the asymptotics of (cid:104)b(n)b(n+h)(cid:105) brings
n≤x
“much the same difficulties” as computing (cid:104)Λ(n)Λ(n+h)(cid:105) , which is a special case of
n≤x
the aforementioned Hardy-Littlewood conjecture.
For triple autocorrelation, as mentioned by Cochrane and Dressler [CD87], it is trivial
that there are infinitely many triples (n−1,n,n+1) with b(n−1)b(n)b(n+1) = 1 (since
one triple like this generates another one, namely (n2 − 1,n2,n2 + 1)) and they give an
upper bound of the expected order of magnitude
1
(cid:104)b(n−1)b(n)b(n+1)(cid:105) (cid:28) .
n≤x (logx)3/2
Hooley [Hoo73] finds infinitely many n with b(n)b(n+h )b(n+h ) = 1 for any h ,h .
1 2 1 2
InthestudyofConnorsandKeating[CK97]onthetwo-pointcorrelationsinthequantum
spectrum of the square billiard, they give a conjectural pair autocorrelation of b at h (cid:54)= 0,
1
(1.7) (cid:104)b(n)b(n+h)(cid:105) ∼ W · , x → ∞
n≤x h logx
where W is an explicit constant depending on h (cf. [Iwa76] for a contradicting conjec-
h
ture in the case h = 1), and they provide numerical data in support of their conjecture.
Assuming (1.7), Smilansky [Smi13] calculates the second moment of the distribution of b
√
in short intervals of length λ logx and shows it is consistent with a Poisson distribution.
Freiberg, Kurlberg, and Rosenzweig [FKR17] give heuristics for higher level autocorrela-
tions and conjecture that for a tuple h = (h ,...,h ) of pairwise distinct integers,
1 k
(1.8) (cid:104)b(n+h )···b(n+h )(cid:105) ∼ S ·(cid:104)b(n)(cid:105)k , x → ∞
1 k n≤x h n≤x
CORRELATIONS IN FUNCTION FIELDS 3
with
(cid:89) δ (p)
h
S = ,
h
δ (p)
0
p
where the product runs over all primes, and
#{n ∈ Z/pνZ : ∀i∃a ,b s.t. n+h ≡ a2 +b2 mod pν}
δ (p) = lim i i i i i .
h ν→∞ pν
It can happen that there is a local obstruction, leading to S = 0, e.g. when the h cover
h i
all residue classes modulo 4, but [FKR17] show that S > 0 if k ≤ 3. We should also note
h
that (1.7) is the special case k = 2 of (1.8), i.e. W = S , see [FKR17, Discussion 1.1].
h (0,h)
Assuming (1.8), [FKR17, Theorem 1.4] deduces that the distribution of the number of
sums of two squares in short intervals of typical length is indeed Poisson. The goal of this
paper is to provide evidence for (1.8) by studying this problem in the function field setting.
1.3. Correlations in the function field setting. In this setting, we replace the ring of
integers by the ring of polynomials F [T] over a finite field F with q elements. The positive
q q
integers up to x are modeled by the subset M ⊆ F [T] of monic polynomials of degree
n,q q
n and the prime polynomials are the monic irreducible polynomials. See e.g. [Rud14] for
the classical analogue of the prime number theorem and a survey of some of the recent
work on number theory in function fields. Our arithmetic functions are complex valued
functions ψ on the monic polynomials M = (cid:83)∞ M . In a general point of view, our
q n=1 n,q
goal is to understand the cross-correlations of arithmetic functions ψ ,...,ψ on M at
1 k q
(h ,...,h ) ∈ F [T]k,
1 k q
(cid:42) (cid:43)
k
(cid:89) 1 (cid:88)
(1.9) ψ (f +h ) = ψ (f +h )···ψ (f +h )
i i qn 1 1 k k
i=1 f∈Mn,q f∈Mn,q
as the parameter qn = #M is large (and n > deg(h ) for all i to avoid technical diffi-
n,q i
culties). This parameter can be large, in particular, either when n is much larger than q,
which we call the large degree limit, or when q is much larger than n, which we call the
large finite field limit.
Typically, in the large degree limit, one knows no more than what is known in number
fields assuming the Generalized Riemann Hypothesis1 (which is, of course, a theorem in
function fields). In the large finite field limit one can often go much further than what can
be done in the number field setting or in the large degree limit. An extensive study by
several authors [ABR15, BB15, Car15, CR14, Pol08] has led to a complete understanding
of (1.9) in this limit for the family of arithmetic functions depending on cycle structure
(see [ABR15, Theorem 1.4]).
1There are, however, several exceptions to this typical phenomenon; see for example, [Hal06, Poo03].
4 LIOR BARY-SOROKER AND ARNO FEHM
1.4. Sums of two squares in the function field setting. There is a recent series of
works on a certain function field analogue of sums of two squares, which we now recall
briefly: For f ∈ M we let
q
(cid:40)
1, if f = A2 +TB2,A,B ∈ F [T]
q
(1.10) b (f) = ,
q
0, otherwise.
√
i.e. we consider norms from the ring F [ −T], which we take as the analogue of Z[i]. With
q
this definition, Smilansky, Wolf and the first named author [BSW16] give asymptotics for
(cid:104)b (f)(cid:105) in the limits q → ∞ and n → ∞, and Gorodetsky [Gor16] extends this to
q f∈Mn,q
(cid:18) (cid:19)
1 2n
(1.11) (cid:104)b (f)(cid:105) ∼ K · , qn → ∞,
q f∈Mn,q q 4n n
where
(cid:89)
(1.12) K = (1−q−1)−1 (1−|P|−2)−1 = 1+O(q−1)
q 2 2
χq(P)=−1
is an explicit constant depending only on q (see Section 2.1 for notation). Moreover, Bank
and the two authors [BBF17] determine the mean value of b in short intervals in the limit
q → ∞. In the spirit of these works, we can formulate a function field analogue of (1.8)
for autocorrelations of b :
q
Conjecture 1.1. Fix N ≥ 1 and k ≥ 1. Then for q an odd prime power, n ≥ N, and
h ,...,h ∈ F [T] of degree less than N and pairwise distinct,
1 k q
(cid:42) (cid:43)
k
(cid:89)
b (f +h ) ∼ S ·(cid:104)b (f)(cid:105)k
q i q,h q f∈Mn,q
i=1 f∈Mn,q
1 (cid:18)2n(cid:19)k
∼ S ·Kk ·
q,h q 4nk n
uniformly as qn → ∞, where K is defined as in (1.12) and
q
(cid:89) δ (P)
q,h
(1.13) S =
q,h δ (P)k
q,0
P∈Fq[T]
monicirred.
with
#{f ∈ F [T]/(Pν) : ∀i∃A ,B f +h ≡ A2 +TB2 mod Pν}
δ (P) = lim q i i i i i .
q,h ν→∞ |P|ν
The constant S comes from the same heuristics that led to the constant S of (1.8),
q,h h
seeSection2fordetails, soConjecture1.1isinperfectanalogywith(1.8). Wenotethatthe
product S always converges and S > 0 if and only if there are no local obstructions
q,h q,h
(Corollary 2.14).
CORRELATIONS IN FUNCTION FIELDS 5
1.5. Results and method. As mentioned before, the main goal of this work is to give
evidence for (1.8) by studying its function field analogue, Conjecture 1.1. We show that
the local factors δ (P) can be computed in theory and we carry out this computation
q,h
in special cases like k = 2. In Section 2.4, we provide numerical evidence that supports
Conjecture 1.1, which is more extensive than what can be done in number fields, as there
is a fast algorithm to compute the analogue of b. We then prove Conjecture 1.1 in the
large finite field limit, where the main term can be explicitly computed:
Theorem 1.2. Fix n ≥ 3 and k ≥ 1. Then for q an odd prime power and h ,...,h ∈ F [T]
1 k q
of degree less than n and pairwise distinct,
(cid:42) (cid:43)
k
(cid:89)
b (f +h ) = S ·(cid:104)b (f)(cid:105)k +O (q−1/2)
q i q,h q f∈Mn,q n,k
i=1 f∈Mn,q
1 (cid:18)2n(cid:19)k
= S · +O (q−1/2),
h 4nk n n,k
where the implied constant depends only on n and k, and
(1.14) S = 2k−#{h1(0),...,hk(0)}.
h
We would like to highlight an interesting phenomenon: In the previously mentioned
results on arithmetic functions that depend only on the cycle structure, it was shown that
in the large finite field limit they become independent; hence the correlation dependence
on h disappears and can be seen only in the error term, cf. [KR16]. However, here, the
dependence on the h is non-trivial and agrees with the heuristics. The simple form of
i
S (as opposed to S ) can be read as saying that in the large finite field limit only the
h q,h
correlation modulo the prime T remains, as f ≡ A2 +TB2 mod T for some A,B ∈ F [T]
q
if and only if f(0) is a square in F .
q
In fact we prove Theorem 1.2 for correlations in short intervals instead of M (see
n,q
Theorem 4.2) and we also phrase Conjecture 1.1 in this generality (see Conjecture 2.2). We
deriveTheorem1.2fromageneralresultoncorrelationsofarithmeticfunctionsthatdepend
on what we call signed factorization type. This result reduces the large finite field limit of
thesecorrelationstocombinatorialproblemsincertainfinitegroups; namely, fiberproducts
of hyperoctahedral groups. In Section 3, we explain this general result (Theorem 3.11) and
its proof, the main part of which consists of showing that the Galois group of a certain
polynomial with a few variable coefficients is a fiber product of hyperoctahedral groups.
Section 4 then contains the combinatorics for b , leading to Theorem 1.2, as well as of a
q
few other arithmetic functions. In particular, we compute (function field analogues of)
the average of b on shifted primes (Theorem 4.7), and autocorrelations of higher divisor
functions twisted by a quadratic character (Theorem 4.12); see the corresponding sections
for more on the history and motivation of these questions.
6 LIOR BARY-SOROKER AND ARNO FEHM
2. Conjectural correlation of sums of two squares
2.1. Preliminaries. With the convention deg0 = −∞, the norm on F [T] is given by
q
|f| = qdegf. A short interval around a polynomial f of degree degf = n is defined
0 0
analogously to the short intervals of integers |n−x| < x(cid:15), cf. [KR14]:
(cid:40) (cid:41)
(cid:88)
I (f ,(cid:15)) := {f ∈ F [T] : |f −f | < |f |(cid:15)} = f + a Ti : a ,...,a ∈ F .
q 0 q 0 0 0 i 0 (cid:98)(cid:15)n(cid:99) q
i<(cid:15)n
So, roughly speaking, f ∈ I (f ,(cid:15)) if and only if 1 − (cid:15) fraction of its higher coefficients
q 0
coincide with those of f .
0
EveryprimepolynomialP ∈ M definesauniqueP-adicvaluationv : F (T) → Z∪{∞}
q P q
with v (P) = 1. The condition that a prime number p is inert in Z[i] corresponds to the
P √
condition that a prime polynomial P ∈ F [T] is irreducible in F [ −T]; equivalently, that
q q
P(−T2) is irreducible in F [T]. Hence, one of the reasons why b is a suitable function
q q
field analogue of b is that it has a multiplicative description very similar to Fermat’s
multiplicative description of b:
Proposition 2.1. Let f ∈ M . Then b (f) = 1 if and only if v (f) ≡ 0 mod 2 for every
q q P
prime polynomial P ∈ F [T] with P(−T2) ∈ F [T] irreducible.
q q
Proof. See [BSW16, Thm. 2.5]. (cid:3)
If we denote by

0, if f(0) = 0,


(2.1) χ : F [T] → {±1,0}, χ (f) = 1, if f(0) ∈ F×2,
q q q q

−1, otherwise.
the quadratic Dirichlet character modulo T, then the condition that P(−T2) is irreducible
is equivalent to χ (P) = −1, see [BSW16, §2].
q
2.2. Heuristics. The notation of this and the following subsection is taken from a pre-
liminary version of [FKR17], and all proofs follow the corresponding proofs there. For
P ∈ F [T] monic irreducible and ν ∈ N we define A (Pν) ⊆ F [T]/(Pν) by
q q q
A (Pν) = {f +(Pν) : f ∈ M ,b (f) = 1}
q q q
= {A2 +TB2 : A,B ∈ F [T]/(Pν)}.
q
Moreover, for h = (h ,...,h ) ∈ F [T]k we let
1 k q
A (Pν) = {f ∈ F [T]/(Pν) : f +h ,...,f +h ∈ A (Pν)}.
q,h q 1 k q
Note that if f ∈ A (Pν) then trivially f ∈ A (Pξ) for all ξ ≤ ν, hence the sequence
q,h q,h
|P|−ν#A (Pν) is monotone decreasing and therefore the limit
q,h
δ (P) = lim |P|−ν#A (Pν)
q,h q,h
ν→∞
exists. Wewillalsoshowbelow(Corollary2.14)thatδ (P) = 0ifandonlyifA (Pν) = ∅
q,h q,h
for some ν > 0 in which case we say that there exists local obstruction at P.
CORRELATIONS IN FUNCTION FIELDS 7
Our heuristic assumption is that the b (f +h ), i = 1,...,k, behave like i.i.d. random
q i
variables as f is randomly picked from a short interval |f −f | < |f |(cid:15), up to a correction
0 0
factor coming from the fact that they are not independent modulo polynomials g. By the
Chinese Remainder Theorem, one may reduce to g = Pν a prime power, for which the
(cid:81)
actual mean of b (f +h ) is given by δ (P) while the random model would predict a
i q i q,h
mean of δ (P)k. This leads to
0
Conjecture 2.2. For every k ≥ 1, d ≥ 1 and 1 ≥ (cid:15) > 0 there exists N ≥ 1 such that for
q an odd prime power, f ∈ F [T] monic of degree n ≥ N and h ,...,h ∈ F [T] of degree
0 q 1 k q
less than d and pairwise distinct,
(cid:42) (cid:43)
k
(cid:89)
(2.2) b (f +h ) ∼ S ·(cid:104)b (f)(cid:105)k
q i q,h q f∈Mn,q
i=1 |f−f0|<|f0|(cid:15)
1 (cid:18)2n(cid:19)k
(2.3) ∼ S ·Kk ·
q,h q 4nk n
uniformly as qn → ∞, where S and K are defined as in (1.13) resp. (1.12).
q,h q
We note that if there exists a local obstruction, then clearly both sides in (2.2) equal 0,
so in this case the conjecture is uninteresting and correct.
2.3. The singular series. We now show that the δ (P) can be computed in theory and
q,h
give estimates in general and concrete formulas in certain special cases. In particular, we
show that the infinite product in (1.13) that defines S indeed converges.
q,h
Lemma 2.3. Let P ∈ F [T] be monic irreducible and f ∈ F [T]. Assume that f (cid:54)≡ 0
q q
mod Pν and write f ≡ Pαg mod Pν with maximal 0 ≤ α < ν and any suitable g ∈ F [T].
q
(1) If χ (P) = −1, then f ∈ A (Pν) if and only if α is even.
q q
(2) If χ (P) = 0, then f ∈ A (Pν) if and only if χ (g) = 1.
q q q
(3) If χ (P) = 1, then f ∈ A (Pν).
q q
Proof. Note that A (Pν) is closed under multiplication and contains P if χ (P) ∈ {0,1},
q q
and P2 if χ (P) = −1. If χ (P) ∈ {±1}, then g ∈ A (Pν): This follows for example
q q q
from the prime polynomial theorem in arithmetic progressions [Ros02] which gives a prime
polynomial Q with Q ≡ g mod Pν and Q ≡ 1 mod T, hence b (Q) = 1 (Proposition 2.1).
q
This proves (3) and the ‘if’ part of (1). For the ‘only if’ part of (1), note that if b (f) = 1
q
and f ≡ Pαg mod Pν, then v (f) = α is even (Proposition 2.1). For the ‘if’ part of (2)
P
we can again apply the prime polynomial theorem to get a prime Q with Q ≡ g mod Pν,
which then satisfies χ (Q) = χ (g) = 1, hence b (Q) = 1. The ‘only if’ part of (2) is
q q q
obvious, since the lowest nonzero coefficient of A2 +TB2 is a square. (cid:3)
Lemma 2.4. Let P ∈ F [T] be monic irreducible.
q
(1) If χ (P) = −1, then
q
(cid:40)
(cid:18) 1 (cid:19) |P| , ν odd
#A (Pν) = |P|ν 1− + |P|+1 .
q |P|+1 1 , ν even
|P|+1
8 LIOR BARY-SOROKER AND ARNO FEHM
(2) If χ (P) = 0, then #A (Pν) = |P|ν+1.
q q 2
(3) If χ (P) = 1, then #A (Pν) = |P|ν.
q q
Proof. This follows by direct counting using Lemma 2.3. For example, if χ (P) = −1 and
q
ν is odd, then the nonzero elements of A (Pν) are represented by polynomials (cid:80)ν−1a Pi
q i=0 i
with deg(a ) < deg(P) and min{i : a (cid:54)= 0} even. Thus,
i i
(cid:88)ν−1 1−|P|−ν−1
#A (Pν)−1 = (|P|−1)|P|ν−α−1 = |P|ν−1(|P|−1) ,
q 1−|P|−2
α=0even
from which the claim follows. (cid:3)
Lemma 2.4 immediately gives δ (P) in the special case h = 0 (which we identify with
q,h
the 1-tuple (0)) or χ (P) = 1:
q
Corollary 2.5. Let P ∈ F [T] monic irreducible. If χ (P) = 1, then δ (P) = 1.
q q q,h
Corollary 2.6. Let P ∈ F [T] monic irreducible. Then
q

1− 1 , χ (P) = −1
 |P|+1 q
δ (P) = 1, χ (P) = 0
q,0 2 q

1, χ (P) = 1.
q
Thecomputationofδ (P)intherestofthecasesismoretechnical. Forh = (h ,...,h )
q,h 1 k
let
(cid:89)
(2.4) ∆ = (h −h ),
h i j
i(cid:54)=j
(2.5) ν (P) = maxv (h −h ).
h P i j
i(cid:54)=j
Lemma 2.7. Let P ∈ F [T] be monic irreducible. Let h = (h ,...,h ) ∈ F [T]k be a
q 1 k q
k-tuple of pairwise distinct polynomials. If χ (P) = −1, then
q
(cid:40)
k |P| , ν odd
|A (Pν)| ≥ |P|ν(1− )+k · |P|+1
q,h |P|+1 1 , ν even
|P|+1
with equality if P (cid:45) ∆ .
h
Proof. WeuseLemma2.4andthatA (Pν) = (cid:84)k A (Pν)and#A (Pν) = #A (Pν),
q,h i=1 q,hi q,hi q
to get that
#A (Pν) = |P|ν −#(F [T]/(Pν)\A (Pν)) ≥ |P|ν −k(|P|ν −#A (Pν)).
q,h q q,h q
If P (cid:45) ∆ , then at most one of f + h ,...,f + h is not in A (Pν) (by Lemma 2.3(1)),
h 1 k q
hence the complements of the A (Pν) are disjoint. (cid:3)
q,hi
Proposition 2.8. If χ (P) = −1, then
q
η
δ (P) = 1−
q,h
|P|+1
where 1 ≤ η ≤ k. Moreover, η = 1 if h = 0, and η = k if P (cid:45) ∆ .
h
CORRELATIONS IN FUNCTION FIELDS 9
Proof. Trivially, A (Pν) ⊆ A (Pν), hence δ (P) ≤ δ (P) is an upper bound, and
q,h q q,h q,0
δ (P) = 1− 1 by Corollary 2.6. The lower bound follows from Lemma 2.7. (cid:3)
q,0 |P|+1
We let
1− k
(cid:89)
˜ |P|+1
S :=
q,k δ (P)k
q,0
χq(P)=−1
and observe that
δ (T) (cid:89) δ (P)
q,h q,h ˜
(2.6) S = · ·S
q,h δ (T)k 1− k q,k
q,0
P|∆h,χq(P)=−1 |P|+1
˜
byProposition2.8andCorollary2.5. SoS isconvergentifandonlyifS isconvergent.
q,h q,k
˜
Lemma 2.9. For every k, the product defining S converges to a positive constant, and
q,k
S˜ = 1+O (q−1).
q,k k
Proof. For each prime polynomial P put
1 k
a = klog(1− )−log(1− ).
P
|P|+1 |P|+1
By Bernoulli’s inequality we have that a ≥ 0 for all P, hence the series
P
(cid:88)
S := a
q P
χq(P)=−1
in convergent if and only if it is bounded. Using the Taylor expansion −log(1 − x) =
x+O(x2), we have that
a (cid:28) |P|−2 = q−2degP.
P
Since the number of P of degree d with χ (P) = −1 is trivially no more than qd, this gives
q
∞
(cid:88) 1
S (cid:28) qdq−2d (cid:28) .
q
q
d=1
˜
Thus the series is convergent and S = O(1/q). Now, as S = exp(−S ), the product
q q,k q
S˜ converges and S˜ = 1+O(1/q). (cid:3)
q,k q,k
From Lemma 2.9 and (2.6) the following assertion immediately follows:
Corollary 2.10. Let h = (h ,...,h ) ∈ F [T]k be a k-tuple of pairwise distinct polynomials
1 k q
and let ∆ as defined in (2.4). Then S converges. Moreover S = 0 if and only if
h q,h q,h
there exists P | T∆ (T) with χ (P) (cid:54)= 1 such that δ (P) = 0.
h q q,h
Next we give convenient formulas for S .
q,h
Lemma 2.11. Let P ∈ F [T] monic irreducible and f ∈ A (Pν).
q q
(1) Assume χ (P) = −1.
q
(a) If f (cid:54)≡ 0 mod Pν or ν is even, then f ∈ A (Pν+1).
q
10 LIOR BARY-SOROKER AND ARNO FEHM
(b) If f ≡ 0 mod Pν and ν is odd, then f ∈ A (Pν+1) if and only if f ≡ 0
q
mod Pν+1.
(2) Assume χ (P) = 0.
q
(a) If f (cid:54)≡ 0 mod Pν, then f ∈ A (Pν+1).
q
(b) If f ≡ 0 mod Pν, then f ≡ αPν mod Pν+1 with α ∈ F , and f ∈ A (Pν+1)
q q
if and only if α = 0 or α ∈ F×2.
q
Proof. This is immediate from Lemma 2.3. (cid:3)
We define
Ω (Pν) = {f ∈ A (Pν) : ∃i f +h ≡ 0 mod Pν},
q,h q,h i
(2.7)
Ω∗ (Pν) = {f ∈ A (Pν) : ∀i f +h (cid:54)≡ 0 mod Pν}.
q,h q,h i
Note that if ν > ν (P), then h +(Pν),...,h +(Pν) are pairwise distinct, so f +h ≡ 0
h 1 k i
mod Pν for at most one i, and
(2.8) #Ω (Pν) = #{i : h −h ∈ A (Pν) for all j}.
q,h j i q
Lemma 2.12. Let P ∈ F [T] be monic irreducible and f ∈ A (Pν), let h = (h ,...,h ) ∈
q q 1 k
F [T]k be a k-tuple of pairwise distinct polynomials, and ν as defined in (2.5).
q h
(1) Assume χ (P) = −1.
q
(a) If f ∈ Ω∗ (Pν), then f ∈ Ω∗ (Pξ) for all ξ ≥ ν.
q,h q,h
(b) If f ∈ Ω (Pν) and ν > ν (P), then f + gPν ∈ Ω (Pν+1) for precisely
q,h h q,h
one g ∈ F [T] with deg(g) < deg(P). For all g(cid:48) (cid:54)= g with deg(g(cid:48)) < deg(P),
q
f +g(cid:48)Pν ∈ Ω∗ (Pν+1) if ν is even, and f +g(cid:48)Pν ∈/ A (Pν+1) if ν is odd.
q,h q,h
(2) Assume χ (P) = 0.
q
(a) If f ∈ Ω∗ (Pν), then f ∈ Ω∗ (Pξ) for all ξ ≥ ν.
q,h q,h
(b) If f ∈ Ω (Pν) and ν > ν (P), then f + αPν ∈ Ω (Pν+1) for one α ∈ F ,
q,h h q,h q
f + αPν ∈ Ω∗ (Pν+1) for q−1 many α ∈ F , and f + αPν ∈/ A (Pν+1) for
q,h 2 q q,h
q−1 many α ∈ F .
2 q
Proof. This follows by applying Lemma 2.11 to the f +h . (cid:3)
i
Proposition 2.13. Let P ∈ F [T] be monic irreducible, let h = (h ,...,h ) ∈ F [T]k be a
q 1 k q
k-tuple of pairwise distinct polynomials, and ν as in (2.5). Fix ν > ν (P).
h h
(1) If χ (P) = −1, then
q
(cid:40)
|P|−ν(#Ω∗ (Pν)+ 1 #Ω (Pν)), ν odd
δ (P) = q,h |P|+1 q,h
q,h |P|−ν(#Ω∗ (Pν)+ |P| #Ω (Pν)), ν even
q,h |P|+1 q,h
(2) If χ (P) = 0, then
q
1
δ (P) = |P|−ν(#Ω∗ (Pν)+ #Ω (Pν)).
q,h q,h 2 q,h