Correlations of sums of two squares and other arithmetic functions in function fields
CCORRELATIONS OF SUMS OF TWO SQUARES AND OTHERARITHMETIC FUNCTIONS IN FUNCTION FIELDS
LIOR BARY-SOROKER AND ARNO FEHM
Abstract.
We investigate a function field analogue of a recent conjecture on autocorre-lations of sums of two squares by Freiberg, Kurlberg and Rosenzweig, which generalizes anolder conjecture by Connors and Keating. In particular, we provide extensive numericalevidence and prove it in the large finite field limit. Our method can also handle correla-tions of other arithmetic functions and we give applications to (function field analoguesof) the average of sums of two squares on shifted primes, and to autocorrelations of higherdivisor functions twisted by a quadratic character. Introduction
We study function field analogues of conjectures on autocorrelations of sums of two squares.1.1.
Correlations of arithmetic functions.
A basic statistical property of an arithmeticfunction ψ is its mean value ; that is, the asymptotic as x → ∞ of(1.1) (cid:104) ψ ( n ) (cid:105) n ≤ x = 1 x (cid:88) n ≤ x ψ ( n ) . More information is given by the cross-correlations of arithmetic functions ψ , . . . , ψ k ,which are defined at ( h , . . . , h k ) ∈ Z k as the asymptotic as x → ∞ of(1.2) (cid:42) k (cid:89) i =1 ψ i ( n + h i ) (cid:43) n ≤ x = 1 x (cid:88) n ≤ x ψ ( n + h ) · · · ψ k ( n + h k ) . In the special case when all the ψ i are equal, we use the term autocorrelations . Someof the most famous theorems and problems in number theory are about these statisticalproperties for certain specific arithmetic functions, like the Hardy-Littlewood prime tupleconjecture, a quantitative version of the twin prime conjecture, which can be expressed interms of the autocorrelations of the von Mangoldt function Λ.1.2. Autocorrelation of sums of two squares.
An integer n is a sum of two squaresif there exist x, y ∈ Z such that n = x + y ; i.e., it is a norm of the Gaussian integer x + iy ∈ Z [ i ]. We let(1.3) b ( n ) = (cid:40) , if there exist x, y ∈ Z such that n = x + y , , otherwise. a r X i v : . [ m a t h . N T ] J a n LIOR BARY-SOROKER AND ARNO FEHM
The study of the statistics of b ( n ) has a long history: Already Landau [Lan08] gives themean value of b as(1.4) (cid:104) b ( n ) (cid:105) n ≤ x = 1 x (cid:88) n ≤ x b ( n ) ∼ K · √ log x , x → ∞ , where(1.5) K = 1 √ (cid:89) p ≡ (1 − p − ) − / ≈ . b in short intervals in many works by variousauthors, including [Hoo74, Iwa76, Hoo94], see the introduction of [BBF17, § b , one has lower and upper bounds of the right order ofmagnitude for the pair autocorrelations:(1.6) 1log x (cid:28) (cid:104) b ( n ) b ( n + h ) (cid:105) n ≤ x (cid:28) x . For h = 1, the upper bound was proved by Rieger [Rie65] while the lower bound byIndlekofer and Schwarz in [IS72, Ind74, Sch72] (see Kelly [Kel78] and Bantle [Ban86] forshort 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) n ≤ x brings“ much the same difficulties ” as computing (cid:104) Λ( n )Λ( n + h ) (cid:105) n ≤ x , which is a special case ofthe aforementioned Hardy-Littlewood conjecture.For triple autocorrelation, as mentioned by Cochrane and Dressler [CD87], it is trivialthat there are infinitely many triples ( n − , n, n + 1) with b ( n − b ( n ) b ( n + 1) = 1 (sinceone triple like this generates another one, namely ( n − , n , n + 1)) and they give anupper bound of the expected order of magnitude (cid:104) b ( n − b ( n ) b ( n + 1) (cid:105) n ≤ x (cid:28) x ) / . Hooley [Hoo73] finds infinitely many n with b ( n ) b ( n + h ) b ( n + h ) = 1 for any h , h .In the study of Connors and Keating [CK97] on the two-point correlations in the quantumspectrum of the square billiard, they give a conjectural pair autocorrelation of b at h (cid:54) = 0,(1.7) (cid:104) b ( n ) b ( n + h ) (cid:105) n ≤ x ∼ W h · x , x → ∞ where W h is an explicit constant depending on h (cf. [Iwa76] for a contradicting conjec-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 √ log x 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 k ) of pairwise distinct integers,(1.8) (cid:104) b ( n + h ) · · · b ( n + h k ) (cid:105) n ≤ x ∼ S h · (cid:104) b ( n ) (cid:105) kn ≤ x , x → ∞ ORRELATIONS IN FUNCTION FIELDS 3 with S h = (cid:89) p δ h ( p ) δ ( p ) , where the product runs over all primes, and δ h ( p ) = lim ν →∞ { n ∈ Z /p ν Z : ∀ i ∃ a i , b i s.t. n + h i ≡ a i + b i mod p ν } p ν . It can happen that there is a local obstruction, leading to S h = 0, e.g. when the h i coverall residue classes modulo 4, but [FKR17] show that S h > k ≤
3. We should also notethat (1.7) is the special case k = 2 of (1.8), i.e. W h = S (0 ,h ) , see [FKR17, Discussion 1.1].Assuming (1.8), [FKR17, Theorem 1.4] deduces that the distribution of the number ofsums of two squares in short intervals of typical length is indeed Poisson. The goal of thispaper 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 ofintegers by the ring of polynomials F q [ T ] over a finite field F q with q elements. The positiveintegers up to x are modeled by the subset M n,q ⊆ F q [ T ] of monic polynomials of degree n and the prime polynomials are the monic irreducible polynomials. See e.g. [Rud14] forthe classical analogue of the prime number theorem and a survey of some of the recentwork on number theory in function fields. Our arithmetic functions are complex valuedfunctions ψ on the monic polynomials M q = (cid:83) ∞ n =1 M n,q . In a general point of view, ourgoal is to understand the cross-correlations of arithmetic functions ψ , . . . , ψ k on M q at( h , . . . , h k ) ∈ F q [ T ] k ,(1.9) (cid:42) k (cid:89) i =1 ψ i ( f + h i ) (cid:43) f ∈ M n,q = 1 q n (cid:88) f ∈ M n,q ψ ( f + h ) · · · ψ k ( f + h k )as the parameter q n = M n,q is large (and n > deg( h i ) for all i to avoid technical diffi-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 numberfields assuming the Generalized Riemann Hypothesis (which is, of course, a theorem infunction fields). In the large finite field limit one can often go much further than what canbe done in the number field setting or in the large degree limit. An extensive study byseveral authors [ABR15, BB15, Car15, CR14, Pol08] has led to a complete understandingof (1.9) in this limit for the family of arithmetic functions depending on cycle structure (see [ABR15, Theorem 1.4]). There are, however, several exceptions to this typical phenomenon; see for example, [Hal06, Poo03].
LIOR BARY-SOROKER AND ARNO FEHM
Sums of two squares in the function field setting.
There is a recent series ofworks on a certain function field analogue of sums of two squares, which we now recallbriefly: For f ∈ M q we let b q ( f ) = (cid:40) , if f = A + T B , A, B ∈ F q [ T ]0 , otherwise. , (1.10)i.e. we consider norms from the ring F q [ √− T ], which we take as the analogue of Z [ i ]. Withthis definition, Smilansky, Wolf and the first named author [BSW16] give asymptotics for (cid:104) b q ( f ) (cid:105) f ∈ M n,q in the limits q → ∞ and n → ∞ , and Gorodetsky [Gor16] extends this to (cid:104) b q ( f ) (cid:105) f ∈ M n,q ∼ K q · n (cid:18) nn (cid:19) , q n → ∞ , (1.11)where(1.12) K q = (1 − q − ) − (cid:89) χ q ( P )= − (1 − | P | − ) − = 1 + O ( q − )is an explicit constant depending only on q (see Section 2.1 for notation). Moreover, Bankand 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 ≥ and k ≥ . Then for q an odd prime power, n ≥ N , and h , . . . , h k ∈ F q [ T ] of degree less than N and pairwise distinct, (cid:42) k (cid:89) i =1 b q ( f + h i ) (cid:43) f ∈ M n,q ∼ S q,h · (cid:104) b q ( f ) (cid:105) kf ∈ M n,q ∼ S q,h · K kq · nk (cid:18) nn (cid:19) k uniformly as q n → ∞ , where K q is defined as in (1.12) and S q,h = (cid:89) P ∈ F q [ T ]monic irred . δ q,h ( P ) δ q, ( P ) k (1.13) with δ q,h ( P ) = lim ν →∞ { f ∈ F q [ T ] / ( P ν ) : ∀ i ∃ A i , B i f + h i ≡ A i + T B i mod P ν }| P | ν . The constant S q,h comes from the same heuristics that led to the constant S h of (1.8),see Section 2 for details, so Conjecture 1.1 is in perfect analogy with (1.8). We note that theproduct S q,h always converges and S q,h > ORRELATIONS IN FUNCTION FIELDS 5
Results and method.
As mentioned before, the main goal of this work is to giveevidence for (1.8) by studying its function field analogue, Conjecture 1.1. We show thatthe local factors δ q,h ( P ) can be computed in theory and we carry out this computationin special cases like k = 2. In Section 2.4, we provide numerical evidence that supportsConjecture 1.1, which is more extensive than what can be done in number fields, as thereis a fast algorithm to compute the analogue of b . We then prove Conjecture 1.1 in thelarge finite field limit, where the main term can be explicitly computed: Theorem 1.2.
Fix n ≥ and k ≥ . Then for q an odd prime power and h , . . . , h k ∈ F q [ T ] of degree less than n and pairwise distinct, (cid:42) k (cid:89) i =1 b q ( f + h i ) (cid:43) f ∈ M n,q = S q,h · (cid:104) b q ( f ) (cid:105) kf ∈ M n,q + O n,k ( q − / )= S h · nk (cid:18) nn (cid:19) k + O n,k ( q − / ) , where the implied constant depends only on n and k , and S h = 2 k − { h (0) ,...,h k (0) } . (1.14)We would like to highlight an interesting phenomenon: In the previously mentionedresults on arithmetic functions that depend only on the cycle structure, it was shown thatin the large finite field limit they become independent; hence the correlation dependenceon h disappears and can be seen only in the error term, cf. [KR16]. However, here, thedependence on the h i is non-trivial and agrees with the heuristics. The simple form of S h (as opposed to S q,h ) can be read as saying that in the large finite field limit only thecorrelation modulo the prime T remains, as f ≡ A + T B mod T for some A, B ∈ F q [ T ]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 n,q (seeTheorem 4.2) and we also phrase Conjecture 1.1 in this generality (see Conjecture 2.2). Wederive Theorem 1.2 from a general result on correlations of arithmetic functions that dependon what we call signed factorization type . This result reduces the large finite field limit ofthese correlations to combinatorial problems in certain finite groups; namely, fiber productsof hyperoctahedral groups. In Section 3, we explain this general result (Theorem 3.11) andits proof, the main part of which consists of showing that the Galois group of a certainpolynomial with a few variable coefficients is a fiber product of hyperoctahedral groups.Section 4 then contains the combinatorics for b q , leading to Theorem 1.2, as well as of afew 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 divisorfunctions twisted by a quadratic character (Theorem 4.12); see the corresponding sectionsfor more on the history and motivation of these questions. LIOR BARY-SOROKER AND ARNO FEHM Conjectural correlation of sums of two squares
Preliminaries.
With the convention deg 0 = −∞ , the norm on F q [ T ] is given by | f | = q deg f . A short interval around a polynomial f of degree deg f = n is definedanalogously to the short intervals of integers | n − x | < x (cid:15) , cf. [KR14]: I q ( f , (cid:15) ) := { f ∈ F q [ T ] : | f − f | < | f | (cid:15) } = (cid:40) f + (cid:88) i<(cid:15)n a i T i : a , . . . , a (cid:98) (cid:15)n (cid:99) ∈ F q (cid:41) . So, roughly speaking, f ∈ I q ( f , (cid:15) ) if and only if 1 − (cid:15) fraction of its higher coefficientscoincide with those of f .Every prime polynomial P ∈ M q defines a unique P -adic valuation v P : F q ( T ) → Z ∪{∞} with v P ( P ) = 1. The condition that a prime number p is inert in Z [ i ] corresponds to thecondition that a prime polynomial P ∈ F q [ T ] is irreducible in F q [ √− T ]; equivalently, that P ( − T ) is irreducible in F q [ T ]. Hence, one of the reasons why b q is a suitable functionfield analogue of b is that it has a multiplicative description very similar to Fermat’smultiplicative description of b : Proposition 2.1.
Let f ∈ M q . Then b q ( f ) = 1 if and only if v P ( f ) ≡ for everyprime polynomial P ∈ F q [ T ] with P ( − T ) ∈ F q [ T ] irreducible.Proof. See [BSW16, Thm. 2.5]. (cid:3)
If we denote by χ q : F q [ T ] → {± , } , χ q ( f ) = , if f (0) = 0 , , if f (0) ∈ F × q , − , otherwise.(2.1)the quadratic Dirichlet character modulo T , then the condition that P ( − T ) is irreducibleis equivalent to χ q ( P ) = −
1, see [BSW16, § 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 q [ T ] monic irreducible and ν ∈ N we define A q ( P ν ) ⊆ F q [ T ] / ( P ν ) by A q ( P ν ) = { f + ( P ν ) : f ∈ M q , b q ( f ) = 1 } = { A + T B : A, B ∈ F q [ T ] / ( P ν ) } . Moreover, for h = ( h , . . . , h k ) ∈ F q [ T ] k we let A q,h ( P ν ) = { f ∈ F q [ T ] / ( P ν ) : f + h , . . . , f + h k ∈ A q ( P ν ) } . Note that if f ∈ A q,h ( P ν ) then trivially f ∈ A q,h ( P ξ ) for all ξ ≤ ν , hence the sequence | P | − ν A q,h ( P ν ) is monotone decreasing and therefore the limit δ q,h ( P ) = lim ν →∞ | P | − ν A q,h ( P ν )exists. We will also show below (Corollary 2.14) that δ q,h ( P ) = 0 if and only if A q,h ( P ν ) = ∅ for some ν > local obstruction at P . ORRELATIONS IN FUNCTION FIELDS 7
Our heuristic assumption is that the b q ( f + h i ), i = 1 , . . . , k , behave like i.i.d. randomvariables as f is randomly picked from a short interval | f − f | < | f | (cid:15) , up to a correctionfactor coming from the fact that they are not independent modulo polynomials g . By theChinese Remainder Theorem, one may reduce to g = P ν a prime power, for which theactual mean of (cid:81) i b q ( f + h i ) is given by δ q,h ( P ) while the random model would predict amean of δ ( P ) k . This leads to Conjecture 2.2.
For every k ≥ , d ≥ and ≥ (cid:15) > there exists N ≥ such that for q an odd prime power, f ∈ F q [ T ] monic of degree n ≥ N and h , . . . , h k ∈ F q [ T ] of degreeless than d and pairwise distinct, (cid:42) k (cid:89) i =1 b q ( f + h i ) (cid:43) | f − f | < | f | (cid:15) ∼ S q,h · (cid:104) b q ( f ) (cid:105) kf ∈ M n,q (2.2) ∼ S q,h · K kq · nk (cid:18) nn (cid:19) k (2.3) uniformly as q n → ∞ , where S q,h and K q are defined as in (1.13) resp. (1.12) . 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 q,h ( P ) can be computed in theory andgive estimates in general and concrete formulas in certain special cases. In particular, weshow that the infinite product in (1.13) that defines S q,h indeed converges. Lemma 2.3.
Let P ∈ F q [ T ] be monic irreducible and f ∈ F q [ T ] . Assume that f (cid:54)≡ P ν and write f ≡ P α g mod P ν with maximal ≤ α < ν and any suitable g ∈ F q [ T ] .(1) If χ q ( P ) = − , then f ∈ A q ( P ν ) if and only if α is even.(2) If χ q ( P ) = 0 , then f ∈ A q ( P ν ) if and only if χ q ( g ) = 1 .(3) If χ q ( P ) = 1 , then f ∈ A q ( P ν ) .Proof. Note that A q ( P ν ) is closed under multiplication and contains P if χ q ( P ) ∈ { , } ,and P if χ q ( P ) = −
1. If χ q ( P ) ∈ {± } , then g ∈ A q ( P ν ): This follows for examplefrom the prime polynomial theorem in arithmetic progressions [Ros02] which gives a primepolynomial Q with Q ≡ g mod P ν and Q ≡ T , hence b q ( Q ) = 1 (Proposition 2.1).This proves (3) and the ‘if’ part of (1). For the ‘only if’ part of (1), note that if b q ( f ) = 1and f ≡ P α g mod P ν , then v P ( f ) = α is even (Proposition 2.1). For the ‘if’ part of (2)we can again apply the prime polynomial theorem to get a prime Q with Q ≡ g mod P ν ,which then satisfies χ q ( Q ) = χ q ( g ) = 1, hence b q ( Q ) = 1. The ‘only if’ part of (2) isobvious, since the lowest nonzero coefficient of A + T B is a square. (cid:3) Lemma 2.4.
Let P ∈ F q [ T ] be monic irreducible.(1) If χ q ( P ) = − , then A q ( P ν ) = | P | ν (cid:18) − | P | + 1 (cid:19) + (cid:40) | P || P | +1 , ν odd | P | +1 , ν even . LIOR BARY-SOROKER AND ARNO FEHM (2) If χ q ( P ) = 0 , then A q ( P ν ) = | P | ν +12 .(3) If χ q ( P ) = 1 , then A q ( P ν ) = | P | ν .Proof. This follows by direct counting using Lemma 2.3. For example, if χ q ( P ) = − ν is odd, then the nonzero elements of A q ( P ν ) are represented by polynomials (cid:80) ν − i =0 a i P i with deg( a i ) < deg( P ) and min { i : a i (cid:54) = 0 } even. Thus, A q ( P ν ) − ν − (cid:88) α =0 e ven ( | P | − | P | ν − α − = | P | ν − ( | P | −
1) 1 − | P | − ν − − | P | − , from which the claim follows. (cid:3) Lemma 2.4 immediately gives δ q,h ( P ) in the special case h = 0 (which we identify withthe 1-tuple (0)) or χ q ( P ) = 1: Corollary 2.5.
Let P ∈ F q [ T ] monic irreducible. If χ q ( P ) = 1 , then δ q,h ( P ) = 1 . Corollary 2.6.
Let P ∈ F q [ T ] monic irreducible. Then δ q, ( P ) = − | P | +1 , χ q ( P ) = − , χ q ( P ) = 01 , χ q ( P ) = 1 . The computation of δ q,h ( P ) in the rest of the cases is more technical. For h = ( h , . . . , h k )let ∆ h = (cid:89) i (cid:54) = j ( h i − h j ) , (2.4) ν h ( P ) = max i (cid:54) = j v P ( h i − h j ) . (2.5) Lemma 2.7.
Let P ∈ F q [ T ] be monic irreducible. Let h = ( h , . . . , h k ) ∈ F q [ T ] k be a k -tuple of pairwise distinct polynomials. If χ q ( P ) = − , then |A q,h ( P ν ) | ≥ | P | ν (1 − k | P | + 1 ) + k · (cid:40) | P || P | +1 , ν odd | P | +1 , ν evenwith equality if P (cid:45) ∆ h .Proof. We use Lemma 2.4 and that A q,h ( P ν ) = (cid:84) ki =1 A q,h i ( P ν ) and A q,h i ( P ν ) = A q ( P ν ),to get that A q,h ( P ν ) = | P | ν − F q [ T ] / ( P ν ) \ A q,h ( P ν )) ≥ | P | ν − k ( | P | ν − A q ( P ν )) . If P (cid:45) ∆ h , then at most one of f + h , . . . , f + h k is not in A q ( P ν ) (by Lemma 2.3(1)),hence the complements of the A q,h i ( P ν ) are disjoint. (cid:3) Proposition 2.8. If χ q ( P ) = − , then δ q,h ( P ) = 1 − η | P | + 1 where ≤ η ≤ k . Moreover, η = 1 if h = 0 , and η = k if P (cid:45) ∆ h . ORRELATIONS IN FUNCTION FIELDS 9
Proof.
Trivially, A q,h ( P ν ) ⊆ A q ( P ν ), hence δ q,h ( P ) ≤ δ q, ( P ) is an upper bound, and δ q, ( P ) = 1 − | P | +1 by Corollary 2.6. The lower bound follows from Lemma 2.7. (cid:3) We let ˜ S q,k := (cid:89) χ q ( P )= − − k | P | +1 δ q, ( P ) k and observe that(2.6) S q,h = δ q,h ( T ) δ q, ( T ) k · (cid:89) P | ∆ h ,χ q ( P )= − δ q,h ( P )1 − k | P | +1 · ˜ S q,k by Proposition 2.8 and Corollary 2.5. So S q,h is convergent if and only if ˜ S q,k is convergent. Lemma 2.9.
For every k , the product defining ˜ S q,k converges to a positive constant, and ˜ S q,k = 1 + O k ( q − ) .Proof. For each prime polynomial P put a P = k log(1 − | P | + 1 ) − log(1 − k | P | + 1 ) . By Bernoulli’s inequality we have that a P ≥ P , hence the series S q := (cid:88) χ q ( P )= − a P in convergent if and only if it is bounded. Using the Taylor expansion − log(1 − x ) = x + O ( x ), we have that a P (cid:28) | P | − = q − P . Since the number of P of degree d with χ q ( P ) = − q d , this gives S q (cid:28) ∞ (cid:88) d =1 q d q − d (cid:28) q . Thus the series is convergent and S q = O (1 /q ). Now, as ˜ S q,k = exp( − S q ), the product˜ S q,k converges and ˜ S q,k = 1 + O (1 /q ). (cid:3) From Lemma 2.9 and (2.6) the following assertion immediately follows:
Corollary 2.10.
Let h = ( h , . . . , h k ) ∈ F q [ T ] k be a k -tuple of pairwise distinct polynomialsand let ∆ h as defined in (2.4) . Then S q,h converges. Moreover S q,h = 0 if and only ifthere exists P | T ∆ h ( T ) with χ q ( P ) (cid:54) = 1 such that δ q,h ( P ) = 0 . Next we give convenient formulas for S q,h . Lemma 2.11.
Let P ∈ F q [ T ] monic irreducible and f ∈ A q ( P ν ) .(1) Assume χ q ( P ) = − .(a) If f (cid:54)≡ P ν or ν is even, then f ∈ A q ( P ν +1 ) . (b) If f ≡ P ν and ν is odd, then f ∈ A q ( P ν +1 ) if and only if f ≡ P ν +1 .(2) Assume χ q ( P ) = 0 .(a) If f (cid:54)≡ P ν , then f ∈ A q ( P ν +1 ) .(b) If f ≡ P ν , then f ≡ αP ν mod P ν +1 with α ∈ F q , and f ∈ A q ( P ν +1 ) if and only if α = 0 or α ∈ F × q .Proof. This is immediate from Lemma 2.3. (cid:3)
We define Ω q,h ( P ν ) = { f ∈ A q,h ( P ν ) : ∃ i f + h i ≡ P ν } , Ω ∗ q,h ( P ν ) = { f ∈ A q,h ( P ν ) : ∀ i f + h i (cid:54)≡ P ν } . (2.7)Note that if ν > ν h ( P ), then h + ( P ν ) , . . . , h k + ( P ν ) are pairwise distinct, so f + h i ≡ P ν for at most one i , and(2.8) q,h ( P ν ) = { i : h j − h i ∈ A q ( P ν ) for all j } . Lemma 2.12.
Let P ∈ F q [ T ] be monic irreducible and f ∈ A q ( P ν ) , let h = ( h , . . . , h k ) ∈ F q [ T ] k be a k -tuple of pairwise distinct polynomials, and ν h as defined in (2.5) .(1) Assume χ q ( P ) = − .(a) If f ∈ Ω ∗ q,h ( P ν ) , then f ∈ Ω ∗ q,h ( P ξ ) for all ξ ≥ ν .(b) If f ∈ Ω q,h ( P ν ) and ν > ν h ( P ) , then f + gP ν ∈ Ω q,h ( P ν +1 ) for preciselyone g ∈ F q [ T ] with deg( g ) < deg( P ) . For all g (cid:48) (cid:54) = g with deg( g (cid:48) ) < deg( P ) , f + g (cid:48) P ν ∈ Ω ∗ q,h ( P ν +1 ) if ν is even, and f + g (cid:48) P ν / ∈ A q,h ( P ν +1 ) if ν is odd.(2) Assume χ q ( P ) = 0 .(a) If f ∈ Ω ∗ q,h ( P ν ) , then f ∈ Ω ∗ q,h ( P ξ ) for all ξ ≥ ν .(b) If f ∈ Ω q,h ( P ν ) and ν > ν h ( P ) , then f + αP ν ∈ Ω q,h ( P ν +1 ) for one α ∈ F q , f + αP ν ∈ Ω ∗ q,h ( P ν +1 ) for q − many α ∈ F q , and f + αP ν / ∈ A q,h ( P ν +1 ) for q − many α ∈ F q .Proof. This follows by applying Lemma 2.11 to the f + h i . (cid:3) Proposition 2.13.
Let P ∈ F q [ T ] be monic irreducible, let h = ( h , . . . , h k ) ∈ F q [ T ] k be a k -tuple of pairwise distinct polynomials, and ν h as in (2.5) . Fix ν > ν h ( P ) .(1) If χ q ( P ) = − , then δ q,h ( P ) = (cid:40) | P | − ν ( ∗ q,h ( P ν ) + | P | +1 q,h ( P ν )) , ν odd | P | − ν ( ∗ q,h ( P ν ) + | P || P | +1 q,h ( P ν )) , ν even(2) If χ q ( P ) = 0 , then δ q,h ( P ) = | P | − ν ( ∗ q,h ( P ν ) + 12 q,h ( P ν )) . ORRELATIONS IN FUNCTION FIELDS 11
Proof. (1): By Lemma 2.12(1), for each ξ ≥ ν , we have q,h ( P ξ ) = q,h ( P ν ) and ∗ q,h ( P ξ +1 ) = | P | · ∗ q,h ( P ξ ) + (cid:40) ( | P | − q,h ( P ξ ) ξ even0 ξ oddso, as ξ → ∞ , | P | − ξ ∗ q,h ( P ξ ) tends to | P | − ν ∗ q,h ( P ν ) + | P | − ν q,h ( P ν ) · ( | P | − · (cid:40)(cid:80) ∞ µ =0 | P | − (2 µ +1) ν even (cid:80) ∞ µ =1 | P | − µ ν odd , from which the claim follows, as | P | − ξ ( A q,h ( P ξ ) − ∗ q,h ( P ξ )) = | P | − ξ q,h ( P ν ) → q,h ( P ν ) ≤ k .(2): By Lemma 2.12(2), for each ξ ≥ ν , we have q,h ( P ξ ) = q,h ( P ν ) and ∗ q,h ( P ξ +1 ) = | P | · ∗ q,h ( P ξ ) + | P | − · q,h ( P ξ ) , so, as ξ → ∞ , | P | − ξ ∗ q,h ( P ξ ) tends to | P | − ν ∗ q,h ( P ν ) + | P | − ν q,h ( P ν ) · | P | − · ∞ (cid:88) µ =1 | P | − µ , from which again the claim follows. (cid:3) Note the striking similarity between Proposition 2.13(2) and the formula for δ k (2) in[FKR17]. As an immediate consequence of Corollary 2.6 and Proposition 2.13 we have Corollary 2.14.
There exists local obstruction at P if and only if δ q,h ( P ) = 0 . In partic-ular, S q,h = 0 if and only if there exists local obstruction at some prime P . We now give explicit formulas for the case k = 2; i.e. h = ( h , h ). In this case, we mayassume that h is of the form h = (0 , h ). Proposition 2.15.
For h = (0 , h ) and P ∈ F q [ T ] monic irreducible with v P ( h ) = ρ wehave δ q,h ( P ) = − | P | ρ +1 | P | ρ ( | P | +1) , χ q ( P ) = − − | P | ρ − | P | ρ +1 , χ q ( P ) = 01 , χ q ( P ) = 1 . In particular, δ q,h ( P ) > for all P , so there exists no local obstruction in the case k = 2 .Proof. Apply Proposition 2.13 with ν = ρ + 1. If χ q ( P ) = − ν is odd, then q,h ( P ν ) = 2 and ∗ q,h ( P ν ) = | P | ν (1 −| P | − ρ )1+ | P | − + | P | −
2. If χ q ( P ) = − ν is even,then q,h ( P ν ) = 0 and ∗ q,h ( P ν ) = | P | ν − | P | − . If χ q ( P ) = 0 and h = T ρ g , then q,h ( P ν ) = 1+ ( χ q ( g )+ χ q ( − g )) and ∗ q,h ( P ν ) = q ν − q + ( q − − χ q ( g ) − χ q ( − g )). (cid:3) Example 2.16.
For h = (0 , δ q,h ( P ) = − | P | +1 , χ q ( P ) = − − | P | , χ q ( P ) = 01 , χ q ( P ) = 1For h = (0 , T ), we get δ q,h ( P ) = − | P | +1 , χ q ( P ) = − − | P | − | P | , χ q ( P ) = 01 , χ q ( P ) = 1In particular, we conclude that S q, (0 ,T ) S q, (0 , = δ q, (0 ,T ) ( T ) δ q, (0 , ( T ) = 2 q − q − q ( q −
1) = 2 + 1 q , so Conjecture 2.2 predicts that(2.9) (cid:104) b q ( f ) b q ( f + T ) (cid:105) f ∈ M n,q (cid:104) b q ( f ) b q ( f + 1) (cid:105) f ∈ M n,q ∼ q , q n → ∞ . Remark 2.17.
We note that, as opposed to the situation in integers, for small q there canbe a local obstruction in the case k = 3, e.g. if q = 3 and h = (0 , , δ q,h ( T ) = 0. Proposition 2.18.
Fix k ≥ and d ≥ . For h , . . . , h k ∈ F q [ T ] pairwise distinct and ofdegree less than d , S q,h = S h + O k,d ( q − / ) , where the implied constant depends only on k and d , and S q,h and S h are defined as in(1.13) and (1.14).Proof. Applying Lemma 2.9, Proposition 2.8, Corollary 2.6 to (2.6) and noting that | P | ≥ q gives S q,h = δ q,h ( T ) δ q, ( T ) k · (cid:89) P | ∆ h ,χ q ( P )= − δ q,h ( P )1 − k | P | +1 · ˜ S q,k = 2 k δ q,h ( T ) + O ( q − ) . Let S = { h (0) , . . . , h k (0) } and l = S . Since on F × q , χ q +12 is the indicator function of A q ( T ), by the definition of Ω ∗ q,h in (2.7) we can write ∗ q,h ( T ) = (cid:88) α ∈ F q − α/ ∈ S (cid:89) β ∈ S χ q ( α + β ) + 12 = 2 − l (cid:88) S (cid:48) ⊆ S (cid:88) α ∈ F q − α/ ∈ S χ q ( (cid:89) β ∈ S (cid:48) ( α + β )) . The Hasse-Weil theorem gives that (cid:80) α ∈ F q − α/ ∈ S χ q ( (cid:81) β ∈ S (cid:48) ( α + β )) = O ( q − / ) when S (cid:48) (cid:54) = ∅ .When S (cid:48) = ∅ , one trivially has (cid:80) α ∈ F q − α/ ∈ S χ q ( (cid:81) β ∈ S (cid:48) ( α + β )) = q − l . Thus, ∗ q,h ( T ) = ORRELATIONS IN FUNCTION FIELDS 13 q − l l + O ( q / ) and q − ν ∗ q,h ( T ν ) = q − ν · ( q ν − ∗ q,h ( T ) + O ( q ν − )) = 12 l + O ( q − / ) . By Proposition 2.13, δ q,h ( T ) = 2 − l + O ( q − / ), as claimed. (cid:3) We conclude this section by pointing out that δ q,h ( P ) could be defined (and possiblycomputed) also in different ways, for example as the asymptotic density of A q,h ( P ∞ ) = { f ∈ F q [ T ] : f + ( P ν ) ∈ A q,h ( P ν ) for all ν } like in [FKR17], or as the Haar measure of a suitable set in the completion of F q [ T ] at P .2.4. Numerics.
In this section we compare Conjecture 2.2 with numerical computations.All our computation were done with the SageMath mathematics software system [Sage].The algorithm to compute the function b q ( f ) is based on factoring f into irreducibles andthen applying Proposition 2.1. Here we exploit the fact that there is a fast factorizationalgorithm for F q [ T ] implemented, as opposed to the situation in integers.In the computations of the singular series, we need to compute δ q,h ( P ), for which weapply Proposition 2.8 and Proposition 2.13, among other results.We enumerate the polynomials in F q [ T ] in the following order: 0 , , . . . , T, T + 1 , . . . ,and we will use this enumeration also in what follows.2.4.1. Varying h . Conjecture 1.1 predicts that for each h ∈ F q [ T ],(2.10) (cid:104) b q ( f ) b q ( f + h ) (cid:105) f ∈ M n,q (cid:104) b q ( f ) b q ( f + 1) (cid:105) f ∈ M n,q ∼ S q, (0 ,h ) S q, (0 , , n → ∞ . In Figures 1 and 2, we compare this conjecture with numerics for n = 100, q = 3 and the729 different h of degree <
6, enumerated as h = 0 , h = 1 , . . . , h = 2 T + · · · + 2 asindicated above. We have sampled N = 5 n = 50000 monic f ∈ F q [ T ] of degree n chosenat random and compute N ( h i ) = (cid:80) b q ( f ) b q ( f + h i ) (cid:80) b q ( f ) b q ( f + 1) and H ( h i ) = S q, (0 ,h i ) S q, (0 , , where the sum in N is taken over the sampled f ’s and H ( h i ) is computed precisely.Figure 2 is analogous to [CK97, Figure 2].2.4.2. Varying n . For h = ( h , . . . , h k ) ∈ F q [ T ] k a k -tuple of pairwise distinct polynomials,Conjecture 1.1 predicts that(2.11) (cid:104) (cid:81) i b q ( f + h i ) (cid:105) f ∈ M n,q (cid:104) b q ( f ) (cid:105) kf ∈ M n,q → S q,h , n → ∞ . We compare this conjecture with numerics for q = 5, 3 ≤ n <
100 and h ∈ { (0 , , (0 , T ) } in Figure 3, and h = (0 , , T ) in Figure 4. For each n we have sampled N = qn / monic Figure 1.
The plots show N ( h i ) (in light grey) and H ( h i ) (in dark grey)as functions of i for the first 31 indices (in the left plot) and for a randomstarting point (in the right plot) Figure 2. N ( h i ) /H ( h i ) as a function of 1 ≤ i ≤ f ∈ F q [ T ] of degree n chosen at random, and for each h = ( h , . . . , h k ) as above wecomputed N ,h ( n ) = N k − (cid:80) (cid:81) ki =1 b q ( f + h i )( (cid:80) b q ( f )) k , where the sums are taken over the sampled f ’s.2.4.3. Varying q.
Conjecture 1.1 (cf. (2.9)) suggests that(2.12) (cid:104) b q ( f ) b q ( f + T ) (cid:105) f ∈ M n,q (cid:104) b q ( f ) b q ( f + 1) (cid:105) f ∈ M n,q = 2 + 1 q (1 + o (1)) , q → ∞ . In Figure 5 we compare this conjecture with numerics. For each prime 2 < q <
30 weevaluate the left hand side of (2.12) by sampling random monic f ∈ F q [ T ] of degrees 3 ORRELATIONS IN FUNCTION FIELDS 15
Figure 3.
The plots of N , (0 , ( n ) (on the left) and N , (0 ,T ) (on the right) asfunctions of n . The lines are approximated values of the respective singularseries S , (0 , and S , (0 ,T ) . Figure 4. N , (0 , ,T ) ( n ) as a function of n . The line is an approximated valueof the respective singular series S , (0 , ,T ) .up to 49 (and for q = 29 up to 69), and this we denote by N ( q ). We compare it to H ( q ) = 2 + 1 /q .2.4.4. Short intervals.
We finally check Conjecture 2.2 about autocorrelations at h = (0 , q = 3 and compute(2.13) N ,f ( (cid:15) ) = (cid:104) b q ( f ) b q ( f + 1) (cid:105) | f − f | < | f | (cid:15) deterministically for fixed f of degree n ≤
150 and various (cid:15) ≤ .
1. Note that N ,f ( (cid:15) ) ispiecewise constant, namely constant on each interval in ≤ (cid:15) < i +1 n , and we chose one epsilonin each of these intervals, namely (cid:15) = in . In Figure 6 we take two f (namely f = T Figure 5.
The curve is H ( q ) = 2 + q as a function of q and the points are( q, N ( q )) for primes 2 < q < Figure 6.
Here we plot N ,f ( (cid:15) ) /H , deg( f ) as a function of (cid:15) , for f ( T ) = T (in the left diagram) and a f of degree 150 that is randomly chosen (inthe right diagram).and a f of the same degree that is randomly chosen ), and we vary (cid:15) . In Figure 7 we take f = T + 2 T + T + T + 2 T + T + T + T + T + 2 T + 2 T + T + T +2 T + 2 T + T + 2 T + 2 T + 2 T + T + T + 2 T + T + T + 2 T + 2 T + T + T + 2 T + 2 T + 2 T + T + T + 2 T + T + 2 T + 2 T + 2 T + T + 2 T + 2 T +2 T + T + 2 T + T + T + 2 T + T + T + T + 2 T + 2 T + T + 2 T + T + 2 T + T + T + 2 T + T + 2 T + T + T + 2 T + 2 T + T + 2 T + T + 2 T + 2 T + 2 T + 2 T + T + T + 2 T + T + 2 T + 2 T + 2 T + 2 T + T + T + T + T + T + 2 T + 2 T +2 T + 2 T + 2 T + 2 T + T + 2 T + T + 2 T + 2 T + 2 T + 2 T + 2 T + 2 ORRELATIONS IN FUNCTION FIELDS 17
Figure 7.
Here we plot N ,T n ( (cid:15) ) as a function of n , for (cid:15) = 0 . (cid:15) = 0 .
05 (in the right diagram), and compare it to H ,n ≈ . /n . (cid:15) = 0 . (cid:15) = 0 .
05, and we vary n . In each case, we compare N ,f ( (cid:15) ) to the value H ,n = S , (0 , · (cid:18) K · n (cid:18) nn (cid:19)(cid:19) ≈ S , (0 , K πn predicted by Conjecture 2.2, where we use approximate values S , (0 , ≈ . K ≈ . n the short interval contains very few polynomials – for examplefor 60 ≤ n <
80 just enough to contain either 0 or 1 element f with b q ( f ) b q ( f + 1) = 1.3. Arithmetic functions of signed factorization type
In this section we define arithmetic functions depending on signed factorization type (seeSection 3.1) and prove a general result (Theorem 3.11) for correlations of such functionsin short intervals. This result explains the autocorrelations through distributions on finitegroups. More precisely, it allows to compute the large finite field limit of an expression ofthe form (cid:42) k (cid:89) i =1 ψ i ( f + h i ) (cid:43) | f − f | < | f | (cid:15) as an average over corresponding functions on a certain finite group depending only on thecombinatorics of h (0) , . . . , h k (0). In Section 3.1 we introduce arithmetic functions depend-ing on signed factorization type, in Section 3.2 we explain the corresponding functions onfinite groups and prove a general statement using a Chebotarev theorem. In Sections 3.3and 3.4 we then study the finite groups that are relevant for correlations in short intervals,namely fiber products of hyperoctahedral groups, and in Section 3.5 we state and provethe general theorem. Signed factorization type.
A signed factorization type is a function λ : N × N × {± , } → Z ≥ with finite support, and we denote by Λ the set of all signed factorization types. For λ ∈ Λwe let deg( λ ) = (cid:88) d ∈ N (cid:88) e ∈ N (cid:88) s ∈{± , } λ ( d, e, s ) de and χ ( λ ) = (cid:40) ( − (cid:80) d,e ∈ N λ ( d,e, − e , if λ ( d, e,
0) = 0 for all d, e , otherwise.(3.1)To each f ∈ M q with prime factorization f = P e · · · P e r r we assign a signed factorizationtype by setting λ f ( d, e, s ) := { i : deg( P i ) = d, e i = e, χ q ( P i ) = s } . Note that deg( λ f ) = deg( f ) ,χ ( λ f ) = χ q ( f ) ,f is square-free ⇔ (cid:80) d,s (cid:80) e> λ f ( d, e, s ) = 0 ,f (0) (cid:54) = 0 ⇔ (cid:80) d,e λ f ( d, e,
0) = 0 . We denote by Λ ∗ the space of functions ψ : Λ → C . Each function ψ ∈ Λ ∗ induces a familyof arithmetic functions ψ q (for q an odd prime power) on M q given by ψ q : M q → C , f (cid:55)→ ψ ( λ f ) . Many arithmetic functions are of this form, in particular:(1) The restriction of the quadratic character χ q (see (2.1)) to M q , which is inducedfrom χ defined in (3.1).(2) The indicator function of prime polynomials, which is induced from1 P ( λ ) = (cid:40) , if (cid:80) d,s λ ( d, , s ) = (cid:80) d,e,s λ ( d, e, s ) = 10 , otherwise , and the closely related function field analogue of the von Mangoldt function, seee.g. [KR14], which is induced fromΛ( λ ) = (cid:40) d , if (cid:80) e,s λ ( d , e, s ) = (cid:80) d,e,s λ ( d, e, s ) = 1 , , otherwise , (3) The function b q defined in (1.10), which by Proposition 2.1 is induced from b ( λ ) = (cid:40) , if λ ( d, e + 1 , −
1) = 0 for all d, e, , otherwise.(4) The function field analogue of the M¨obius function µ , see e.g. [CR14]. ORRELATIONS IN FUNCTION FIELDS 19 (5) The function field analogue of the function r counting the number of representationsas sums of two squares, see Section 4.1.(6) The function field analogue of the r -divisor function d r , see Section 4.3.3.2. Arithmetic functions on hyperoctahedral groups.
The hyperoctahedral group (aka, signed permutation group ) of degree n is the permutational wreath product(3.2) H n = F (cid:111) S n = V (cid:111) S n with V = ( F ) n . The S n -invariant subspaces of V are(3.3) V n = V, V n − = { x ∈ V : (cid:80) ni =1 x i = 0 } ,V = { (0 , . . . , , (1 , . . . , } , V = { (0 , . . . , } , cf. [BBF17, Lemma 4.2]. In particular, V n − (cid:111) S n , V n (cid:111) A n and V n − (cid:111) A n are normalsubgroups of H n . The total sign homomorphism(3.4) χ n : H n → {± } is given by χ n ( xτ ) = ( − (cid:80) i x i , where x = ( x , . . . , x n ) ∈ V and τ ∈ S n . It is obvious that χ n is surjective and ker χ n = V n − (cid:111) S n .To each σ = xτ ∈ H n , with x ∈ V and τ ∈ S n , we assign a signed factorization type bysetting(3.5) λ σ ( d, e, s ) = (cid:40) { Ω : Ω is an orbit of τ, d, ( − (cid:80) i ∈ Ω x i = s } , if e = 10 , otherwiseNote that deg( λ σ ) = n,χ ( λ σ ) = χ n ( σ ) . Each ψ ∈ Λ ∗ induces a family of maps ψ n : H n → C , g (cid:55)→ ψ ( λ g ) . In particular, the function χ defined in (3.1) induces the total sign maps χ n . From nowon, we abuse notation and write χ instead of χ n .We denote by H kn the direct product of k copies of H n and by π i : H kn → H n the pro-jection onto the i -th component. For a subgroup G of H kn we define a k -multilinear map (cid:104)·(cid:105) G : (Λ ∗ ) k → C by (cid:104) ψ , . . . , ψ k (cid:105) G := (cid:42) k (cid:89) i =1 ψ i,n ( π i ( σ )) (cid:43) σ ∈ G = 1 G · (cid:88) σ ∈ G k (cid:89) i =1 ψ i ( λ π i ( σ ) ) . For example, if G = H kn , we have independence ; namely, (cid:104) ψ , . . . , ψ k (cid:105) G = k (cid:89) i =1 (cid:104) ψ i (cid:105) H n . Now we describe the connection with arithmetic. Let K be a field of characteris-tic different from 2 and let f ∈ K [ T ] of degree n . Suppose that f is separable and f (0) (cid:54) = 0. If ω , . . . , ω n is an enumeration of the roots of f , then the roots of f ( − T )are ± ρ , . . . , ± ρ n , where ρ i is a fixed square root of − ω i . The Galois action then inducesembeddings Gal( f ( T ) | K ) → S n and(3.6) Θ : Gal( f ( − T ) | K ) → H n = V (cid:111) S n , where the elements of S n act by permuting the ω i , and the elements of V act by sign changeon the ρ i , cf. [BBF17, Lemma 3.1]. Over the finite K = F q , the Galois group in (3.6) isgenerated by the Frobenius automorphism φ q : x (cid:55)→ x q . In fact λ f and λ Θ( φ q ) are equal: Lemma 3.1.
Let f ∈ F q [ T ] monic, square-free and not divisible by T . Then λ f = λ Θ( φ q ) ,where φ q is the Frobenius automorphism and Θ is the map from (3.6) Proof.
Let f = P e · · · P e r r be the prime factorization of f . Note that by assumption, χ q ( P i ) ∈ {± } and e i = 1 for all i , hence λ f ( d, e, s ) = 0 = λ Θ( φ ) ( d, e, s ) for all ( d, e, s ) witheither e > s = 0. The set of roots Ω = { ω , . . . , ω n } of f is partitioned as Ω = (cid:96) ri =1 Ω i ,where Ω i is the set of roots of P i . Write Θ( φ q ) = xτ , and note that the Ω i are exactly theorbits of τ , of length i = deg( P i ). By [BBF17, Lemma 3.3], χ q ( P i ) = ( − (cid:80) j ∈ Ω i x j . Itfollows that for any ( d, s ) with s (cid:54) = 0, λ f ( d, , s ) = { i : deg( P i ) = d, e i = 1 , χ q ( P i ) = s } = { i : i = d, ( − (cid:80) j ∈ Ω i x j = s } = λ Θ( φ ) ( d, , s ) . Thus, λ f ( d, e, s ) = λ Θ( φ ) ( d, e, s ) for all ( d, e, s ). (cid:3) Given k polynomials f , . . . , f k ∈ K [ T ] that are of degree n , separable, with f i (0) (cid:54) =0, and pairwise coprime, the Galois group Gal( (cid:81) ki =1 f i ( − T ) | K ) embeds into H kn . Thefollowing proposition computes the limit of correlations of arithmetic functions dependingon signed factorization type, where the correlation can be taken over any set of polynomialsin F q [ T ] that are specializations of a fixed polynomial f A ∈ F q [ A , . . . , A m ][ T ], whereby a specialization of f A we mean the polynomial f a ∈ F q [ T ] obtained by substituting a , . . . , a m ∈ F q for the variables A , . . . , A m . For example, the short interval | f − f | < q m is the set of specializations of f A = f + (cid:80) m − i =0 A i T i . Proposition 3.2.
Fix n ≥ , m ≥ , k ≥ and ψ , . . . , ψ k ∈ Λ ∗ . Let q be an odd primepower, let f ,A , . . . , f k,A ∈ F q [ A , . . . , A m ][ T ] be monic, of degree n , square-free, not divisibleby T , and pairwise coprime, and denote g A ( T ) = k (cid:89) i =1 f i,A ( − T ) and G = Gal( g A | F q ( A )) ≤ H kn . Assume that G = Gal( g A | F q ( A )) . Then (cid:42) k (cid:89) i =1 ψ i,q ( f i,a ) (cid:43) a ∈ F mq = (cid:104) ψ , . . . , ψ k (cid:105) G + O ( q − / ) , ORRELATIONS IN FUNCTION FIELDS 21 where the implied constant depends only on n , m , k and ψ , . . . , ψ k .Proof. For λ = ( λ , . . . , λ k ) ∈ Λ k let A ( λ ) = { a ∈ F mq : ( λ f ,a , . . . , λ f k,a ) = λ } and G ( λ ) = { σ ∈ G : ( λ π ( σ ) , . . . , λ π k ( σ ) ) = λ } , and note that k (cid:89) i =1 ψ i,q ( f i,a ) = k (cid:89) i =1 ψ i ( λ i ) = k (cid:89) i =1 ψ i,n ( π i ( σ ))for all a ∈ A ( λ ) and σ ∈ G ( λ ). Thus, since each ψ ∈ Λ ∗ takes only a finite number ofvalues on the set of λ ∈ Λ with deg( λ ) = n , it suffices to prove for each λ that A ( λ ) q m = G ( λ ) G + O ( q − / ) . Note that the assumption on the f i,A ’s can be rephrased as saying that the discriminant ∆ ∈ F q [ A ] of g A is non-zero. Also note that ∆( a ) is the discriminant of g a ( T ) = (cid:81) ki =1 f i,a ( − T ).We distinguish two cases:If λ i ( d, e, s ) > i and either e > s = 0, then λ π i ( σ ) equals λ i for no σ , so G ( λ ) = 0. On the other side, λ f i,a = λ i only if ∆( a ) = 0, so as deg(∆) is bounded onlyin terms of n and k , we get A ( λ ) = O ( q m − ), see e.g. [Sch76, Ch. 4 Lemma 3A].If λ i ( d, e, s ) = 0 for all i and all ( d, e, s ) with e > s = 0, then we have embed-dings Θ i,a : Gal( f i,a | F q ) → H n and Θ a : Gal( (cid:81) i f i,a | F q ) → H kn , and both Galois groups aregenerated by the Frobenius automorphism φ q . Applying the Chebotarev theorem [BBF17,Theorem 5.1] to the polynomial g A and the G -invariant set G ( λ ) gives that { a ∈ F mq : ∆( a ) (cid:54) = 0 and Θ a ( φ q ) ∈ G ( λ ) } = G ( λ ) G · q m + O ( q m − / ) . (3.7)Since Θ i,a ( φ q ) = π i (Θ a ( φ q )), and λ f i,a = λ Θ i,a ( φ q ) by Lemma 3.1, we get that the left handside of (3 .
7) equals precisely A ( λ ), which gives the claim. (cid:3) Next we provide some group theoretical tools to determine G in the case f i,A = f + h i + (cid:80) mj =0 A j T j , needed to apply Proposition 3.2 for correlations in short intervals.3.3. Fiber products of hyperoctahedral groups.
For a finite group G we denote by M ( G ) the intersection over the maximal normal subgroups of G . In the literature, M ( G )is sometimes called the Melnikov subgroup of G . It is the smallest normal subgroup of G with quotient a direct product of simple groups. We will use several times that subgroupsof finite elementary abelian p -groups are again elementary abelian p -groups and thereforeproducts of simple groups. We will also use that if N (cid:2) G , then M ( N ) ⊆ M ( G ), and if f : G → H is a homomorphism, then f ( M ( G )) = M ( f ( G )), cf. [FJ08, Lemma 25.5.4]. Lemma 3.3.
Let ( G i ) i ∈ I be a family of finite groups and let G := (cid:81) i ∈ I G i . Then M ( G ) = (cid:81) i ∈ I M ( G i ) . Proof.
For each i , G i (cid:2) G implies M ( G i ) ⊆ M ( G ), and therefore (cid:81) i ∈ I M ( G i ) ⊆ M ( G ).The other inclusion follows as G/ (cid:81) i ∈ I M ( G i ) = (cid:81) i ∈ I G i /M ( G i ) is a product of simplegroups, so M ( G ) = (cid:81) i ∈ I M ( G i ). (cid:3) Proposition 3.4.
Let ( G i ) i ∈ I be a finite family of finite groups, put G = (cid:81) i ∈ I G i , let H ≤ G , and let π i : G → G i be the projection map. If HM ( G ) = G and π i ( H ) = G i forall i , then H = G .Proof. Suppose first that I = { , } . Then by Goursat’s lemma, there exist a group Q andepimorphisms f i : G i → Q such that H is the fiber product G × Q G . Let M be a maximalnormal subgroup of Q , put S = Q/M , and let π : Q → S be the quotient map. Then H is contained in the fiber product G × S G . Since M i := ker( π ◦ f i ) is a maximal normalsubgroup of G i , we have M ( G ) = M ( G ) × M ( G ) ⊆ M × M ⊆ G × S G . Thus, HM ( G ) ⊆ G × S G (cid:36) G × G , contradicting the assumption HM ( G ) = G .The general case now follows by induction: Indeed, for J ⊆ I , put G J = (cid:81) i ∈ J G j , let π J : G → G J be the projection map, and let H J := π J ( H ). Then π i ( H J ) = π i ( H ) = G i forall i ∈ J , and H J M ( G J ) = π J ( H ) M ( π J ( G )) ⊇ π J ( HM ( G )) = π J ( G ) = G J , so if J = I \ { i } , then the induction hypothesis can be applied to G J = (cid:81) i ∈ J G i and H J ,showing that H J = G J , and then the case I = { , } can be applied to G = G J × G i and H , showing that H = G . (cid:3) Next we compute the Melnikov subgroups of the hyperoctahedral group and of one ofits subgroups.
Lemma 3.5.
Under the notation of § n ≥ , then M ( V (cid:111) S n ) = M ( V n − (cid:111) S n ) = V n − (cid:111) A n . Proof.
Let W be either V n or V n − . Write H = W (cid:111) S n and K = V n − (cid:111) A n ≤ H. It is immediate that M ( H ) ⊆ K , as K is either a maximal normal subgroup of H , if W = V n − or an intersection of such, if W = V . To show the converse inclusion, we let N be a maximal normal subgroup of H and aim to show that K ⊆ N . Let A := H/N and U := N ∩ W . Then A is a finite simple group and U is an S n -invariant subspace of V thatis contained in W .If N does not contain W , then W N = H from maximality. Thus, W/U ∼ = H/N = A .As W is a 2-elementary abelian group, this implies that A ∼ = Z / Z and that U has index2 in W . By (3.3), since n ≥ W = V and U = V n − . Hence, frommaximality N = V n − (cid:111) S n and thus K ⊆ N . ORRELATIONS IN FUNCTION FIELDS 23 If N contains W , then N/W is a maximal normal subgroup of
H/W ∼ = S n . Since S n hasa unique maximal normal subgroup, namely A n , we conclude that N = W (cid:111) A n and so N contains K . (cid:3) For a finite set I we let H ( I ) n be the fiber product of copies of H n with respect to χ ; i.e.,H ( I ) n := (cid:40) ( g i ) i ∈ I ∈ (cid:89) i ∈ I H n : χ ( g i ) = χ ( g j ) for all i, j ∈ I (cid:41) , and we denote by π i : H ( I ) n → H n the i -projection map. Note that (cid:89) i ∈ I V n (cid:111) S n ≥ H ( I ) n ≥ (cid:89) i ∈ I V n − (cid:111) S n ≥ (cid:89) i ∈ I V n − (cid:111) A n . Lemma 3.6.
For n ≥ and any finite set I , M ( (cid:89) i ∈ I V n (cid:111) S n ) = M (H ( I ) n ) = M ( (cid:89) i ∈ I V n − (cid:111) S n ) = (cid:89) i ∈ I V n − (cid:111) A n . Proof.
Let(3.8) Γ = (cid:89) i ∈ I V n (cid:111) S n , N = (cid:89) i ∈ I V n − (cid:111) S n , and K = (cid:89) i ∈ I V n − (cid:111) A n . As N is normal in G with abelian quotient and χ ( x ) = 1 for each x ∈ V n − (cid:111) S n , we havethat N (cid:2) H ( I ) n (cid:2) Γ. Hence M ( N ) ≤ M (H ( I ) n ) ≤ M (Γ). Lemmas 3.3 and 3.5 give that M (Γ) = M ( N ) = K , which thus proves the claim. (cid:3) We prove an analogue of Proposition 3.4 for fiber products of H n . Proposition 3.7.
Fix a finite set I , put G = H ( I ) n , and let H ≤ G . If HM ( G ) = G and π i ( H ) = H n for all i ∈ I , then H = G .Proof. Let N and K be as in (3.8). By Lemma 3.6, M ( G ) = M ( N ) = K . The assumption HK = HM ( G ) = G implies an isomorphism G/K ∼ = H/H ∩ K . In particular, if we put H = H ∩ N , then as N is a normal in G , H is normal in H with H/H ∼ = G/N ∼ = F andwe have N = H K .Since π i ( H ) = π i ( G ) and [ π i ( G ) : π i ( N )] = 2, we conclude that π i ( H ) = π i ( N ).Therefore, Proposition 3.4 gives that H = N , from which we conclude that H = G . (cid:3) We shall need the following notation for later use: For a finite family I = ( I j ) j ∈ J ofnonempty finite sets we denoteH I n := (cid:89) j ∈ J H ( I j ) n ≤ (cid:89) j ∈ J (cid:89) i ∈ I j H n . The Galois group of correlations in short intervals.
Let F be a field of char-acteristic different from 2, let n > m ≥ r >
0, and let K = F ( A , . . . , A m ). Let f ∈ F [ T ] be monic of degree n and h , . . . , h k ∈ F [ T ] of degree less than n and pairwisedistinct. Starting from f ( T ) = f ( T ) + m (cid:88) i =0 A i T i ∈ K [ T ] , we define f i ( T ) = f ( T ) + h i ( T ) and f h ( T ) = (cid:81) ki =1 f i ( T ). Note that f i (0) = f (0) + A + h i (0) . The goal of this section is to determine the Galois group of f h ( − T ).If ω , . . . , ω n is an enumeration of the roots of f i , then the roots of f i ( − T ) are ± ρ , . . . , ± ρ n ,where ρ i is a fixed square root of − ω i . The Galois action induces embeddings Gal( f i ( T ) | K ) (cid:44) → S n and Θ i : Gal( f i ( − T ) | K ) (cid:44) → H n = V (cid:111) S n as in (3.8). Proposition 3.8.
For each i , the action on the roots of f i ( T ) and Θ i induce isomorphisms Gal( f i ( T ) | K ) ∼ = S n and Gal( f i ( − T ) | K ) ∼ = H n = V (cid:111) S n . Here, the extensions of K corresponding to the subgroups V (cid:111) A n and V n − (cid:111) S n of H n are K ( (cid:112) discr( f i )) and K ( (cid:112) f i (0)) , respectively.Proof. See [BBF17, Proposition 4.6] for the claimed isomorphisms. The subgroup V (cid:111) A n of V (cid:111) S n corresponds to the same extension as the subgroup A n of S n , namely to K ( (cid:112) discr( f i )), cf. [Mil14, Corollary 4.2]. Since f i (0) = ( − n ω · · · ω n = ( ρ · · · ρ n ) , thesplitting field of f i ( − T ) contains K ( (cid:112) f i (0)). Since the elements of V n − act on the roots of f i ( − T ) by an even number of sign changes, (cid:112) f i (0) = ρ · · · ρ n is invariant under V n − (cid:111) S n ,which implies that K ( (cid:112) f i (0)) is the field extension of K corresponding to the subgroup V n − (cid:111) S n of H n . (cid:3) As before we have the mapsΨ i : Gal( f h ( − T ) | K ) res i (cid:47) (cid:47) Gal( f i ( − T ) | K ) Θ i (cid:47) (cid:47) H n , i = 1 , . . . , k, which induce an embedding(3.9) Ψ : Gal( f h ( − T ) | K ) → H kn . Denote by H the image of Ψ, G = (cid:81) ki =1 H n , and π i : G → H n the projection onto the i -thfactor. Proposition 3.8 implies that(3.10) π i ( H ) = π i ( G ) = H n , for all i = 1 , . . . , k .Let J = { h i (0) : i = 1 , . . . , k } , (3.11) I a = { i : h i (0) = a } , for a ∈ J, (3.12) I h = ( I a ) a ∈ J . (3.13) ORRELATIONS IN FUNCTION FIELDS 25
Recall that a family ( x , . . . , x s ) of elements of K × is square-independent if the residuesin K × /K × are F -independent. By Kummer theory, this is equivalent toGal( K ( √ x , . . . , √ x s ) | K ) ∼ = ( F ) s . Lemma 3.9.
The family ( f (0) + A + a ) a ∈ J ∪ (discr( f i )) i =1 ,...,k is square-independent in K .Proof. By [BB15, Proposition 2.1], the discr( f i ) are non-squares and pairwise coprime inthe ring R = F ( A , . . . , A m )[ A ]. For a ∈ J and i = 1 , . . . , k , writing f i as ˜ f i + A with A = f (0)+ A + a and ˜ f i ∈ F ( A , . . . , A m )[ T ], then ˜ f i is separable and [BBF17, Lemma 4.5]gives that A does not divide discr( f i ) in R . Since A is a prime element in the UFD R ,for each a ∈ J , this shows that the family ( f (0) + A + a ) a ∈ J ∪ (discr( f i )) i =1 ,...,r consistsof pairwise coprime non-squares in R , which implies that it is square-independent in thefraction field K of R . (cid:3) For each i , let K i = K ( (cid:112) f i (0)) = K ( (cid:112) f (0) + A + h i (0)) and letres K i : Gal( f i ( − T ) | K ) → Gal( K i | K )denote the corresponding restriction map. Proposition 3.10.
The map Ψ given in (3.9) induces an isomorphism Gal( f h ( − T ) | K ) ∼ = H I h n , where I h is defined in (3.13) .Proof. Let H be the image of Gal( f h ( − T ) | K ) in G = (cid:81) ki =1 H n = (cid:81) a ∈ J (cid:81) i ∈ I a H n . As Ψ isinjective, it suffices to show that H = H I h n .We first treat the case J = { a } . Since K i is the same for all i ∈ I a , we call it K a . It isa common subfield of the splitting fields of all f i ( − T ), hence H is contained in H ( I a ) n Weapply Proposition 3.7, where π i ( H ) = H n by (3.10). By Lemma 3.6, M (H ( I a ) n ) = (cid:89) i ∈ I a V n − (cid:111) A n , and M (H ( I a ) n ) ∩ H corresponds to the field extension K a · K ( (cid:112) discr( f i ) , i ∈ I a ) . By Lemma 3.9, f (0) + A + a and (discr( f i )) i ∈ I a are square-independent, which meansthat HM (H ( I a ) n ) = H ( I a ) n . Therefore, we verified the assumptions of Proposition 3.7 and sowe conclude that H = H ( I a ) n , as needed.For general J , if π a (cid:48) : (cid:81) a ∈ J (cid:81) i ∈ I a H a → (cid:81) i ∈ I a (cid:48) H n denotes the projection map, then π a ( H ) = H ( I a ) n for all a ∈ J by the previous case, hence H ≤ (cid:81) a ∈ J H ( I a ) n = H I h n . By Lemma 3.3, M (H I h n ) = (cid:89) a ∈ J M (H ( I a ) n ) = (cid:89) a ∈ J (cid:89) i ∈ I a V n − (cid:111) A n , and M (H I h n ) ∩ H corresponds to the field extension (cid:89) a ∈ J K a · K ( (cid:112) discr( f i ) , i = 1 , . . . , k ) . By Lemma 3.9, the family( f (0) + A + a ) a ∈ J ∪ (discr( f i )) i =1 ,...,k is square-independent, which means that HM (H I h n ) = H I h n . Therefore, Proposition 3.4,applied to the groups (H ( I a ) n ) a ∈ J and the subgroup H ≤ H I h n = (cid:81) a ∈ J H ( I a ) n , gives that H = H I h n . (cid:3) Correlations of arithmetic functions depending on signed factorization type.
We are now ready to prove our general result on correlations in short intervals:
Theorem 3.11.
Fix k ≥ , ψ , . . . , ψ k ∈ Λ ∗ , ≥ (cid:15) > and n > (cid:15) − . Then for q an oddprime power, f ∈ F q [ T ] monic of degree n and h , . . . , h k ∈ F q [ T ] of degree less than n and pairwise distinct, (cid:42) k (cid:89) i =1 ψ i,q ( f + h i ) (cid:43) | f − f | < | f | (cid:15) = (cid:104) ψ , . . . , ψ k (cid:105) H I hn + O n,k,ψ ( q − / ) where the implied constant depends only on n , k and ψ , . . . , ψ k , and I h is defined as in (3 . .Proof. The short interval | f − f | < | f | (cid:15) is precisely the set of specializations of f A = f + (cid:80) m − i =0 A i T i , with m = (cid:100) (cid:15)n (cid:101) . Setting f A,i = f A + h i and f A,h = (cid:81) ki =1 f A,i , Proposition 3.10gives that Gal( f A,h ( − T ) | F q ( A )) = Gal( f A,h ( − T ) | F q ( A )) = H I h n . In particular, the f A,i are monic, square-free, not divisible by T and pairwise coprime.Therefore, the claim follows from Proposition 3.2. (cid:3) We note that in the case where the ψ i depend only on cycle type , i.e. ψ i ( d, ,
1) = ψ i ( d, , −
1) for all d , (cid:104) ψ , . . . , ψ k (cid:105) H I hn = k (cid:89) i =1 (cid:104) ψ i (cid:105) H n , as we will explain in more detail in Lemma 4.8, and hence (cid:42) k (cid:89) i =1 ψ i,q ( f + h i ) (cid:43) f ∈ M n,q = k (cid:89) i =1 (cid:104) ψ i,q ( f ) (cid:105) f ∈ M n,q + O ( q − / ) , which was proven already in [ABR15] (although not stated in such generality). ORRELATIONS IN FUNCTION FIELDS 27 Correlations in the large finite field limit
We now prove Theorem 1.2 in short intervals and compute the correlations of somefurther examples. Due to Theorem 3.11 all that is left to do is to compute the correspondingaverages (cid:104) ψ , . . . , ψ k (cid:105) H I hn , which is a purely combinatorial task.We will use on several occasions the (trivial) principles that if G , G are finite groupsand ψ i : G i → C , then (cid:104) ψ ( σ ) ψ ( σ ) (cid:105) ( σ ,σ ) ∈ G × G = (cid:104) ψ ( σ ) (cid:105) σ ∈ G · (cid:104) ψ ( σ ) (cid:105) σ ∈ G (4.1)and if π : G → G is an epimorphism, then (cid:104) ψ ( π ( σ )) (cid:105) σ ∈ G = (cid:104) ψ ( σ ) (cid:105) σ ∈ G . (4.2)4.1. Autocorrelation of b and r . We will start with the autocorrelation of b , and thenlook at the closely related arithmetic function r , which counts the number of representationsas a sum of two squares. The following general consideration will simplify our arguments: Lemma 4.1.
Let ψ , . . . , ψ k ∈ Λ ∗ and let I = ( I j ) j ∈ J be a partition of { , . . . , k } , i.e. I j (cid:54) = ∅ for all j and { , . . . , k } = (cid:96) j ∈ J I j . If ψ , . . . , ψ k are all supported on χ − (1) = { λ ∈ Λ : χ ( λ ) = 1 } , then (cid:104) ψ , . . . , ψ k (cid:105) H I n = S I · (cid:104) ψ (cid:105) H n · · · (cid:104) ψ k (cid:105) H n , where S I = 2 k − J .Proof. Since by assumption ψ i,n ( σ ) = 0 for σ ∈ H n \ V n − (cid:111) S n and since H I n contains (cid:81) i V n − (cid:111) S n , we get by (4.1) that (cid:104) ψ , . . . , ψ k (cid:105) H I n = 1 I n (cid:88) σ =( σ i ) ∈ (cid:81) i V n − (cid:111) S n (cid:89) i ψ i ( σ i ) = V n − (cid:111) S n ) k I n · k (cid:89) i =1 (cid:104) ψ i (cid:105) V n − (cid:111) S n . This finishes the proof as S I = kn I n . (cid:3) The following result gives the autocorrelations of b q in short intervals. Theorem 1.2 isthe special case with parameters (cid:15) = 1 and f = T n : Theorem 4.2.
Fix k ≥ , ≥ (cid:15) > and n > (cid:15) − . Then for q an odd prime power, f ∈ F q [ T ] monic of degree n and h , . . . , h k ∈ F q [ T ] of degree less than n and pairwisedistinct, (cid:42) k (cid:89) i =1 b q ( f + h i ) (cid:43) | f − f | < | f | (cid:15) = S q,h · (cid:104) b q ( f ) (cid:105) kf ∈ M n,q + O n,k ( q − / )= S h · nk (cid:18) nn (cid:19) k + O n,k ( q − / ) where the implied constant depends only on n and k , and S q,h and S h are defined as in(1.13) and (1.14). Proof.
Set ψ = · · · = ψ k = b and I = I h and note that Lemma 4.1 applies, leading to (cid:104) b, . . . , b (cid:105) H I hn = S h · (cid:104) b (cid:105) k H n , since S I h = S h . Ewens’ sampling formula gives that (cid:104) b (cid:105) H n = n (cid:0) nn (cid:1) , see [BBF17, Lemma 2.3]. Therefore Theorem 3.11 shows that (cid:42) k (cid:89) i =1 b q ( f + h i ) (cid:43) | f − f | < | f | (cid:15) = S h · nk (cid:18) nn (cid:19) k + O n,k ( q − / ) . In particular, (cid:104) b q ( f ) (cid:105) f ∈ M n,q = n (cid:0) nn (cid:1) + O ( q − / ) (which also follows from [BSW16]). To-gether with S h = S q,h + O ( q − / ) (Proposition 2.18), we conclude that S h · nk (cid:18) nn (cid:19) k = S q,h · (cid:104) b q ( f ) (cid:105) kf ∈ M n,q + O n,k ( q − / ) , as needed. (cid:3) Just like the autocorrelation of b (see the introduction), the autocorrelation of r has beenstudied by various authors, but the latter one turns out to be much more accessible: Al-ready Estermann [Est32] proves an asymptotic formula for (cid:104) r ( n ) r ( n + h ) (cid:105) n ≤ x . Apparentlyunaware of that, Connors and Keating [CK97, eqn. (27)] provide a conjectural formulaand numerics for lim x →∞ (cid:104) r ( n ) r ( n + h ) (cid:105) n ≤ x and observe that here the data seems to matchtheir prediction even better than in the case of b . We define the function field analogue of r as r q ( f ) = { ( A, B ) : f = A + T B , A, B ∈ F q [ T ] } / {± } . We note that r q is multiplicative and therefore one obtains, just like for r , the formula r q ( P e · · · P e r r ) = (cid:40)(cid:81) χ q ( P i )=1 ( e i + 1) , if e i is even for all i with χ q ( P i ) = − , otherwise , which implies that r q is induced from r ∈ Λ ∗ defined by(4.3) r ( λ ) = (cid:40)(cid:81) d,e ( e + 1) λ ( d,e, , if λ ( d, e + 1 , −
1) = 0 for all d, e , otherwise . We remark without proof that, like for integers, also the formula r q ( f ) = (cid:88) d | f,d ∈ M q χ q ( d )holds.We start by computing the mean value of r on H n . Lemma 4.3. (cid:104) r (cid:105) H n = 1 Proof.
As usual we write an element σ ∈ H n as σ = xτ with τ ∈ S n and x = ( x , . . . , x n ) ∈ V = F n and we recall the corresponding signed factorization type λ σ attached to σ and given in (3.5). Note that λ σ ( d, e, s ) = 0 if e >
1. Thus, if (cid:80) d λ σ ( d, , −
1) = 0,
ORRELATIONS IN FUNCTION FIELDS 29 then (cid:80) d,e λ σ ( d, e + 1 , −
1) = 0, (cid:81) d,e ( e + 1) λ σ ( d,e, = 2 (cid:80) d λ σ ( d, , , and (cid:80) d λ σ ( d, ,
1) = (cid:80) d,s λ σ ( d, , s ) = (cid:80) d λ τ ( d, , r n ( σ ) = (cid:40) (cid:80) d λ τ ( d, , , if (cid:80) d λ σ ( d, , −
1) = 0 and σ = xτ , otherwise.Let N τ denote the number of σ = xτ ∈ H n with (cid:80) d λ σ ( d, , −
1) = 0. Note that σ iscounted by N τ if and only if for each orbit I ⊂ { , . . . , n } of τ we have (cid:80) i ∈ I x i = 0, andso there are 2 n − j such σ , where j = (cid:88) d,s λ τ ( d, , s ) = (cid:88) d λ τ ( d, , τ . So(4.5) N τ = 2 n − (cid:80) d λ τ ( d, , . By (4.4) and (4.5), we conclude that (cid:88) σ ∈ H n r n ( σ ) = (cid:88) τ ∈ S n (cid:88) x ∈ F n r n ( xτ ) = (cid:88) τ ∈ S n N τ r n ( τ ) = (cid:88) τ ∈ S n n = n !2 n = n , and so (cid:104) r (cid:105) H n = 1. (cid:3) Now we can compute the autocorrelation of r : Theorem 4.4.
Fix k ≥ , ≥ (cid:15) > and n > (cid:15) − . Then for q an odd prime power, f ∈ F q [ T ] monic of degree n and h , . . . , h k ∈ F q [ T ] of degree less than n and pairwisedistinct, (cid:42) k (cid:89) i =1 r q ( f + h i ) (cid:43) | f − f | < | f | (cid:15) = S q,h · (cid:104) r q ( f ) (cid:105) kf ∈ M n,q + O n,k ( q − / )= S h + O n,k ( q − / ) where the implied constant depends only on n and k , and S q,h and S h are defined as in(1.13) and (1.14).Proof. Apply Theorem 3.11 with ψ = · · · = ψ k = r . Note that Lemma 4.1 applies again,so the claim follow from Lemma 4.3. (cid:3) Cross-correlations of b and r with P . We now turn to cross-correlations of b and r with the prime indicator function 1 P , which also have been studied by various authors:Starting from a conjecture of Hardy and Littlewood [HL24, Conjecture J] on the numberof representations of an integer as the sum of two squares and a prime, [Hoo57, Theorem2] proves an asymptotic formula for1 x (cid:88) p ≤ x prime r ( p + h ) = (cid:104) r ( n )1 P ( n − h ) (cid:105) n ≤ x under the Extended Riemann Hypothesis, which [Bre62] then proves unconditionally. Mo-tohashi in [Mot70, Conjecture J*, Theorem 2] and [Mot71] gives a conjectural asymptoticformula for 1 x (cid:88) p ≤ x prime b ( p −
1) = (cid:104) b ( n )1 P ( n + 1) (cid:105) n ≤ x − and proves upper and lower bounds. Iwaniec [Iwa72, p. 204] also proves lower and upperbounds and suggests a correction of Motohashi’s conjecture. We now give a function fieldversion of the Motohashi–Iwaniec conjecture (or rather a generalization of it from h = 1to arbitrary h ) and prove it in the large finite field limit.The heuristics for the cross-correlation of b and 1 P is very similar to that of the auto-correlation of b discussed in Section 2, hence we omit some of the details, and leave themas an exercise for the reader. Let h ∈ F q [ T ] be non-zero. For each prime polynomial P , welet ρ q,h ( P ) = lim ν →∞ { f mod P ν : f ∈ M q prime and ∃ A, B : f + h ≡ A + T B mod P ν }| P | ν , (4.6) ρ q ( P ) = lim ν →∞ { ( f, g ) mod P ν : f ∈ M q prime and ∃ A, B : g ≡ A + T B mod P ν }| P | ν . (4.7)One may verify that the limits indeed exist. So ρ q,h ( P ) /ρ q ( P ) measures the local deviationat P from the random model. One then may make the following Conjecture 4.5.
Fix N ≥ . Then for q an odd prime power, n ≥ N , and nonzero h ∈ F q [ T ] of degree less than N , (cid:104) P ( f ) b q ( f + h ) (cid:105) f ∈ M n,q ∼ T q,h · (cid:104) P ( f ) (cid:105) f ∈ M n,q · (cid:104) b q ( f ) (cid:105) f ∈ M n,q ∼ T q,h · K q · n n (cid:18) nn (cid:19) , uniformly as q n → ∞ , where K q is defined as in (1.12) and (4.8) T q,h = (cid:89) P ∈ F q [ T ]monic irred . ρ q,h ( P ) ρ q ( P ) . We note that T q,h converges. Next we give formulas for the ρ ’s. Note that f is congruentto a prime modulo P ν if and only if either f (cid:54)≡ P or f ≡ P mod P ν , and thereforereplacing the condition “ f prime” by “ f (cid:54)≡ P ” leads to the same limits and at thesame time simplifies the computations. Lemma 4.6.
Let P ∈ F q [ T ] monic irreducible.(1) ρ q ( P ) = (1 − | P | − ) δ q, ( P ) (Recall that δ q, ( P ) is explicitly given by Corollary 2.6.)(2) If χ ( P ) = 1 , then ρ q,h ( P ) = ρ q ( P ) = 1 − | P | − . ORRELATIONS IN FUNCTION FIELDS 31 (3) If χ ( P ) = − , then ρ q,h ( P ) = (cid:40) − | P | − , if P | h − | P | − − (1 + | P | ) − , otherwise.(4) If χ ( P ) = 0 , then ρ q,h ( P ) = 12 − q (1 + χ q ( h )) . Proof. (1) is immediate and (2) follows directly from Lemma 2.3(3) while (3) and (4) followsfrom Lemma 2.3 using similar arguments as those used in the proof of Proposition 2.15. (cid:3)
Just like we deduced Proposition 2.18 from Proposition 2.8 and Corollary 2.6, we canuse Lemma 4.6 to conclude that T q,h = 1 + O ( q − )where the implied constant depends only on the degree of h . Since also K q = 1 + O ( q − ),the next results proves Conjecture 4.5 in the large finite field limit. Theorem 4.7.
Fix ≥ (cid:15) > and n > /(cid:15) . Then for q an odd prime power, f ∈ F q [ T ] monic of degree n and h ∈ F q [ T ] of degree less than n , (cid:104) P ( f ) b q ( f + h ) (cid:105) | f − f | < | f | (cid:15) = T q,h · (cid:104) P ( f ) (cid:105) f ∈ M n,q · (cid:104) b q ( f ) (cid:105) f ∈ M n,q + O n ( q − / )= 1 n n (cid:18) nn (cid:19) + O n ( q − / ) where T q,h is defined as in (4.8) and the implied constant depends only on n .Proof. Apply Theorem 3.11 with k = 2, h = 0, h = − h , ψ = b , ψ = 1 P . Note that ψ satisfies the assumption of the following lemma, which proves the first equality. Forthe second equality use that T q,h = 1 + O ( q − ) and (cid:104) b (cid:105) H n = n (cid:0) nn (cid:1) (see above), and that (cid:104) P (cid:105) H n = n , as this is the fraction of n -cycles in S n . (cid:3) Lemma 4.8.
Let ψ , . . . , ψ k ∈ Λ ∗ and let I = ( I j ) j ∈ J be a partition of { , . . . , k } as above.If ψ k depends only on cycle type in the sense that ψ k ( d, ,
1) = ψ k ( d, , − for all d , then (cid:104) ψ , . . . , ψ k (cid:105) H I n = (cid:104) ψ , . . . , ψ k − (cid:105) H I(cid:48) n · (cid:104) ψ k (cid:105) H n , where I (cid:48) = ( I (cid:48) j ) j ∈ J (cid:48) , I (cid:48) j = I j \ { k } , J (cid:48) = { j ∈ J : I (cid:48) j (cid:54) = ∅} .Proof. Let π : H n → S n be the quotient map. The assumption implies that ψ k,n = ˜ ψ k,n ◦ π with a function ˜ ψ k,n : S n → C . Observe that restricting the homomorphismid H I(cid:48) n × π : H I (cid:48) n × H n → H I (cid:48) n × S n gives an epimorphism H I n → H I (cid:48) n × S n . Thus, (cid:104) ψ , . . . , ψ k (cid:105) H I n (4 . = (cid:42) k − (cid:89) i =1 ψ i,n ( π i ( σ )) · ˜ ψ k,n ( π k ( σ )) (cid:43) σ ∈ H I(cid:48) n × S n (4 . = (cid:42) k − (cid:89) i =1 ψ i,n ( π i ( σ )) (cid:43) σ ∈ H I(cid:48) n · (cid:68) ˜ ψ k,n ( σ ) (cid:69) σ ∈ S n (4 . = (cid:104) ψ , . . . , ψ k − (cid:105) H I(cid:48) n · (cid:104) ψ k (cid:105) H n . (cid:3) Note that although in Theorem 4.7 the functions b q and 1 P become independent in thelarge finite field limit, this could not have been deduced from the earlier results in [ABR15](see the remark after Theorem 3.11), as only one of the two arithmetic functions, namely1 P , depends only on cycle type, while the other one does not.4.3. Autocorrelations of d r χ . Let d r ( n ) be the number of ways to write n as a productof r positive integers. In particular, d = τ is the usual divisor function. The prob-lem of estimating the autocorrelations of d r , sometimes referred to as ‘additive divisorproblem’, ‘shifted divisor’, or ‘shifted convolution’, is well studied both in number fields(see e.g. [CG01, KGH07, CK16] and the recent survey [Tao16]) and function fields (see[ABR15]). The asymptotic of the cross-correlations of the divisor functions are related tocomputing the moments of the zeta function on the critical line, see [Ivi97]. We considera twisted version of this; namely, we twist d r by a quadratic character χ and study thecross-correlations of the d r χ . This is closely related to the moments of the corresponding L -function L ( s, χ ) on the critical line, but we do not elaborate on this any further, sincethe goal of this section is to provide yet another application of Theorem 3.11.In the function field setting, we let d r,q ( f ) denote the number of ways to write f as aproduct of r monic polynomials, and we twist d r,q by χ q , the quadratic character modulo T . We note that in this setting, computing the moments of the corresponding L-function L q ( s, χ q ) is trivial, since L q ( s, χ ) is identically equal to 1, but the cross-correlations of the d r,q χ q are nevertheless interesting and, to the best of our knowledge, unknown. We nowcompute these cross-correlations in the large finite field limit, which we can do as d r,q isinduced from d r ∈ Λ ∗ given by d r ( λ ) = (cid:89) d,e,s (cid:18) e + r − r − (cid:19) λ ( d,e,s ) . Lemma 4.9.
For every r and n , (cid:104) d r (cid:105) H n = (cid:18) n + r − r − (cid:19) . ORRELATIONS IN FUNCTION FIELDS 33
Proof.
Since d r,n factors through π : H n → S n , we have (cid:104) d r (cid:105) H n = 12 n n ! (cid:88) σ ∈ H n (cid:18) r − r − (cid:19) (cid:80) d,s λ σ ( d, ,s ) = 1 n ! (cid:88) τ ∈ S n r ω ( τ ) where ω ( τ ) is the number of cycles of τ . Viewing r ω ( τ ) as the number of partitions of { , . . . , n } into r sets that are unions of orbits of τ and changing order of summation weget that (cid:88) τ ∈ S n r ω ( τ ) = (cid:88) { ,...,n } = (cid:96) ri =1 A i r (cid:89) i =1 A i !Splitting the sum according to the cardinality of A and applying induction on r we con-clude (cid:88) { ,...,n } = (cid:96) ri =1 A i r (cid:89) i =1 A i ! = n (cid:88) a =0 (cid:16) na (cid:17) a !( n − a )! (cid:18) n − a + r − r − (cid:19) = n ! n (cid:88) a =0 (cid:18) a + r − r − (cid:19) = n ! (cid:18) n + r − r − (cid:19) . (cid:3) Lemma 4.10.
Let ψ ∈ Λ ∗ with ψ ( d, ,
1) = ψ ( d, , − for all d . Then (cid:104) ψχ (cid:105) H n = 0 .Proof. Write ψ n = ˜ ψ n ◦ π with π : H n → S n the quotient map. Then (cid:88) σ ∈ H n ψ n ( σ ) χ n ( σ ) = (cid:88) τ ∈ S n (cid:88) σ ∈ π − ( τ ) ψ n ( σ ) χ n ( σ ) = (cid:88) τ ∈ S n ˜ ψ n ( τ ) (cid:88) σ ∈ π − ( τ ) χ n ( σ ) = 0 , since χ n ( σ ) = 1 for half of the 2 n many σ in each π − ( τ ), and χ n ( σ ) = − (cid:3) Lemma 4.11.
Let ψ , . . . , ψ k ∈ Λ ∗ and let I = ( I j ) j ∈ J be a partition of { , . . . , k } . If ψ i ( d, ,
1) = ψ i ( d, , − for all i and d , then (cid:104) ψ χ, . . . , ψ k χ (cid:105) H I n = (cid:40) (cid:104) ψ (cid:105) H n · · · (cid:104) ψ k (cid:105) H n , if I j is even for all j ∈ J , otherwise . Proof.
First observe that since H I n = (cid:81) j ∈ J H ( I j ) n , by principle (4.1) it suffices to prove theclaim in the case J = 1, which we therefore assume now. Note that (cid:104) ψ i χ (cid:105) H n = 0 byLemma 4.10 and thus (cid:104) ψ i ( χ + 1) (cid:105) H n = (cid:104) ψ i (cid:105) H n . So since ψ i ( χ + 1) satisfies the assumptions of Lemma 4.1 and ψ i satisfies the assumptions of Lemma 4.8, we get (cid:104) ψ χ, . . . , ψ k χ (cid:105) H I n = (cid:104) ψ ( χ + 1) − ψ , . . . , ψ k ( χ + 1) − ψ k (cid:105) H I n = k (cid:89) i =1 (cid:104)− ψ i (cid:105) H n + (cid:88) ∅(cid:54) = S ⊆{ ,...,k } (cid:32)(cid:89) i/ ∈ S (cid:104)− ψ i (cid:105) H n · S − (cid:89) i ∈ S (cid:104) ψ i ( χ + 1) (cid:105) H n (cid:33) = ( − k · k (cid:89) i =1 (cid:104) ψ i (cid:105) H n · (cid:88) ∅(cid:54) = S ⊆{ ,...,k } ( − S S − . Now note that (cid:88) S ⊆{ ,...,k } ( − S S = (cid:88) S ⊆{ ,...,k } (cid:88) S ⊆ S ( − S = (cid:88) S ⊆{ ,...,k } (cid:88) S ⊆ S ⊆{ ,...,k } ( − S and (cid:80) S ⊆ S ⊆{ ,...,k } ( − S equals 0 except if S = { , . . . , k } , in which case it equals ( − k .Thus, (cid:104) ψ χ, . . . , ψ k χ (cid:105) H I n = ( − k · k (cid:89) i =1 (cid:104) ψ i (cid:105) H n · (1 + 12 (( − k − , from which the claim follows. (cid:3) Theorem 4.12.
Fix k ≥ , r , . . . , r k ≥ , ≥ (cid:15) > and n > (cid:15) − . Then for q an oddprime power, f ∈ F q [ T ] monic of degree n and h , . . . , h k ∈ F q [ T ] of degree less than n and pairwise distinct, (cid:42) k (cid:89) i =1 d r i ,q χ q ( f + h i ) (cid:43) | f − f | < | f | (cid:15) = D h · k (cid:89) i =1 (cid:104) d r i ,q ( f ) (cid:105) f ∈ M n,q + O ( q − / )= D h · k (cid:89) i =1 (cid:18) n + r i − r i − (cid:19) + O ( q − / ) where the implied constant depends only on n and r , . . . , r k , and D h = (cid:40) , if { i : h i (0) = a } is even for all a ∈ F q , otherwiseProof. Apply Theorem 3.11 with ψ i = d r i χ . Note that d r , . . . , d r k satisfy the assumptionsof Lemma 4.11, and their averages are given by Lemma 4.9. (cid:3) Acknowledgements
The authors are very grateful to Ofir Gorodetsky for sharing his signed factorizationtype viewpoint on sums of two squares with them. They would also like to thank EfratBank for helpful discussions on the topic of this work, Hung Bui for suggesting to studycorrelations relating to moments of L -functions, Jon Keating and Zeev Rudnick for their ORRELATIONS IN FUNCTION FIELDS 35 advice on a preliminary version of the paper, and Tristan Freiberg, P¨ar Kurlberg and LiorRosenzweig for making their preliminary manuscript available to them.The first author is partially supported by the Israel Science Foundation (grant No.925/14), the second author by a research grant from the Ministerium f¨ur Wissenschaft,Forschung und Kunst Baden-W¨urttemberg.
References [ABR15] J. C. Andrade, Lior Bary-Soroker, and Zeev Rudnick. Shifted convolution and the Titchmarshdivisor problem over F q [ t ]. Philosophical Transactions of the Royal Society of London A: Math-ematical, Physical and Engineering Sciences , Theo Murphy meeting issue ‘Number fields andfunction fields: coalescences, contrasts and emerging applications’ compiled and edited by J. P.Keating, Z. Rudnick and T. D. Wooley, 373(2040), 2015. 1.3, 3.5, 4.2, 4.3[BB15] E. Bank and L. Bary-Soroker. Prime polynomial values of linear functions in short intervals.
J.Number Theory
American J. Math. , 2017. 1.2, 1.4, 3.2, 3.2, 3.2, 3.2, 3.4, 3.4,4.1[Ban86] G. Bantle. An asymptotic formula for B -twins. Acta Arithmetica
Finite Fields Appl.
Dokl. Akad. Nauk SSSR
Philosophical Transactions of the Royal Society of London A:Mathematical, Physical and Engineering Sciences , Theo Murphy meeting issue ‘Number fieldsand function fields: coalescences, contrasts and emerging applications’ compiled and edited byJ. P. Keating, Z. Rudnick and T. D. Wooley, 373(2040), 2015. 1.3[CR14] D. Carmon and Z. Rudnick. The autocorrelation of the M¨obius function and Chowla’s conjecturefor the rational function field.
Q. J. Math.
Archiv Math.
J. Phys.A: Math. Gen.
Duke Math. J.
Res. NumberTheory
2, 2016. 4.3[Est32] T. Estermann. An asymptotic formula in the theory of numbers.
Proc. London Math. Soc.
Field Arithmetic . Third Edition. Springer, 2008. 3.3[FKR17] T. Freiberg, P. Kurlberg and L. Rosenzweig. Poisson distribution for gaps between sums of twosquares and level spacings for toral point scatterers. arXiv:1701.01157 [math-ph], 2017. 1.2, 1.2,2.2, 2.3, 2.3[Gor16] O. Gorodetsky. A Polynomial Analogue of Landau’s Theorem and Related Problems.arXiv:1603.02890 [math.NT], 2016. 1.4[Hal06] C. Hall. L -functions of twisted Legendre curves. J. Number Theory [HL24] G. H. Hardy and J. E. Littlewood. Some Problems of ’Partitio Numerorum’(V): A FurtherContribution to the Study of Goldbach’s Problem.
Proc. London Math. Soc.
S2-22 no. 1, 46–56,1924. 4.2[Hoo57] C. Hooley. On the representation of a number as the sum of two squares and a prime.
Acta Math.
Acta Math.
J. Number Theory
J. Reine Angew.Math.
J. Reine Angew.Math.
Archiv Math.
Acta Arith-metica
New trends inprobability and statistics ϕ ( x, y )+ A where ϕ is a quadratic form. Acta Arithmetica
Acta Arithmetica
Duke Math. J
Int. Math.Res. Not.
IMRN 2016, no. 3, 860–874, 2016. 1.5[KR14] J. P. Keating and Z. Rudnick. The variance of the number of prime polynomials in short intervalsand in residue classes.
Int. Math. Res. Not.
IMRN 2014(1):259–288, 2014. 2.1, 2[Kel78] P. J. Kelly. The number of B-twins in an interval. Dissertation, Nottingham, 1978. 1.2[Lan08] E. Landau. ¨Uber die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahlder zu ihrer additiven Zusammensetzung erforderlichen Quadrate.
Arch. Math. Phys.
Fields and Galois theory . Lecture notes, version 4.50, 2014. 3.4[Mot70] Y. Motohashi. On the distribution of prime numbers which are of the form x + y + 1. ActaArithmetica
XVI:351–363, 1970. 4.2[Mot71] Y. Motohashi. On the distribution of prime numbers which are of the form “ x + y + 1”. II. Acta Math. Acad. Sci. Hungar.
Proc. Amer.Math. Soc.
Duke Math. J.
Indag. Math.
Number Theory in Function Fields . Springer, 2002. 2.3[Rud14] Z. Rudnick. Some problems in analytic number theory for polynomials over a finite field.
Pro-ceedings of the ICM vol 1 , 2014. 1.3[Sage] SageMath, the Sage Mathematics Software System (Version SageMath-7.2.beta0), The SageDevelopers, 2016, . 2.4[Sch76] W. M. Schmidt.
Equations over Finite Fields. An Elementary Approach.
Springer 1976. 3.2[Sch72] W. Schwarz. ¨Uber B -Zwillinge II. Archiv Math.
ORRELATIONS IN FUNCTION FIELDS 37 [Smi13] Y. Smilansky. Sums of two squares - pair correlation & distribution in short intervals.
Int. J.Number Theory
09, 2013. 1.2[Tao16] T. Tao. Heuristic computation of correlations of higher order divisor functions.
WordPress.com ,What’s new, Online Blog. http://goo.gl/GBncWw
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