## Abstract

In this paper, we solve a function field analogue of classical problems in analytic number theory, concerning the autocorrelations of divisor functions, in the limit of a large finite field.

## 1. Introduction

The goal of this paper is to study a function field analogue of classical problems in analytic number theory, concerning the autocorrelations of divisor functions. First, we review the problems over the integers and then we proceed to investigate the same problems over the rational function field .

### (a) The additive divisor problem over

Let *d*_{k}(*n*) be the number of representations of *n* as a product of *k* positive integers (*d*_{2} is the standard divisor function). Several authors have studied the *additive divisor problem* (other names are âshifted divisorâ and âshifted convolutionâ), which is to get bounds, or asymptotics, for the sum
1.1
where *h*â 0 is fixed for this discussion.

The case *k*=2 (the ordinary divisor function) has a long history: Ingham [1] computed the leading term, and Estermann [2] gave an asymptotic expansion
1.2
where
1.3
with
1.4
and *a*_{1}(*h*) and *a*_{2}(*h*) are very complicated coefficients.

The size of the remainder term has great importance in applications for various problems in analytic number theory, in particular, the dependence on *h*. See Deshouillers & Iwaniec [3] and Heath-Brown [4] for an improvement of the remainder term.

The higher divisor problem *k*â„3 is also of importance, in particular, in relation to computing the moments of the Riemann *Î¶*-function on the critical line [5,6]. It is conjectured that
1.5
where *P*_{2(kâ1)}(*u*;*h*) is a polynomial in *u* of degree 2(*k*â1), whose coefficients depend on *h* (and *k*). We can get good upper bounds on the additive divisor problem from results in sieve theory on sums of multiplicative functions evaluated at polynomials, for instance, such as those by Nair & Tenenbaum [7]. The conclusion is that for *h*â 0
1.6
and we believe this is the right order of magnitude. But even a conjectural description of the polynomials *P*_{2(kâ1)}(*u*;*h*) is difficult to obtain (see Â§7, [5,6]).

A variant of the problem about the autocorrelation of the divisor function is to determine an asymptotic for the more general sum given by
1.7
Asymptotics are known for the case (*k*,*r*)=(*k*,2) for any positive integer *k*â„2: Linnik [8] showed
1.8

Motohashi [9,10,11] gave an asymptotic expansion
1.9
for all *Î”*>0, where the coefficients *f*_{k,j}(*h*) can in principle be explicitly computed. For an improvement in the *O* term, see Fouvry & Tenenbaum [12].

### (b) The Titchmarsh divisor problem over

A different problem involving the mean value of the divisor function is the *Titchmarsh divisor problem*. The problem is to understand the average behaviour of the number of divisors of a shifted prime, that is, the asymptotics of the sum over primes
1.10
where *a*â 0 is a fixed integer, and . Assuming the generalized Riemann hypothesis (GRH), Titchmarsh [13] showed that
1.11
with
1.12
and this was proved unconditionally by Linnik [8].

Fouvry [14] and Bombieri *et al.* [15] gave a secondary term,
1.13
for all *A*>1 and
1.14
with *Îł* being the EulerâMascheroni constant and Li(*x*) the logarithmic integral function.

In the following sections, we study the additive divisor problem and the Titchmarsh divisor problem over , obtaining definitive analogues of the conjectures described above.

### (c) The additive divisor problem over

We denote by the set of monic polynomials in of degree *n*. Note that .

The divisor function *d*_{k}(â*f*) is the number of ways to write a monic polynomial *f* as a product of *k* monic polynomials:
1.15
where it is allowed to have *a*_{i}=1.

The mean value of *d*_{k}(â*f*) has an exact formula (see lemma 2.2):
1.16
Note that is a polynomial in *n* of degree *k*â1 and leading coefficient 1/(*k*â1)! Our first goal is to study the autocorrelation of *d*_{k} in the limit . We show:

### Theorem 1.1

*Fix n>1. Then*
1.17
*uniformly for all* *of degree* *as* .

In light of (1.16), theorem 1.1 may be interpreted as the statement that *d*_{k}(â*f*) and *d*_{k}(*f*+*h*) become independent in the limit as long as .

To compare with conjecture (1.5) over , we note that is a polynomial in *n* of degree 2(*k*â1) with leading coefficient 1/[(*k*â1)!]^{2}, in agreement with the conjecture (see Â§7b).

The case *h*=0: As an aside, we note that the case *h*=0 is of course dramatically different. Indeed one can show that
1.18
is a polynomial of degree *k*^{2}â1 in *n*, rather than degree 2(*k*â1) for non-zero shifts.

Our method in fact gives the more general result:

### Theorem 1.2

*Let* *k**=(k*_{1}*,âŠ,k*_{s}*) be a tuple of positive integers and* *h**=(h*_{1}*,âŠ,h*_{s}*) a tuple of distinct polynomials in* *. We let*
*Then, for fixed n>1,*
*uniformly on all tuples* *h**=(h*_{1}*,h*_{2}*,âŠ,h*_{s}*) of distinct polynomials in* *of degrees* *as* .

In particular, for **k**=(2,*k*) we get
1.19
in agreement with (1.8).

### (d) The Titchmarsh divisor problem over

Let be the set of monic irreducible polynomials in of degree *n*. By the Prime Polynomial Theorem, we have
Our next result is a solution of the Titchmarsh divisor problem over in the limit of large finite field.

### Theorem 1.3

*Fix n>1. Then*
1.20
*uniformly over all* *of degree* .

For the standard divisor function (*k*=2), we find
1.21
which is analogous to (1.13) under the correspondence and .

### (e) Independence of cycle structure of shifted polynomials

We conclude the introduction with a discussion on the connection between shifted polynomials and random permutations and state a result that lies behind the results stated above.

The cycle structure of a permutation *Ï* of *n* letters is the partition Î»(*Ï*)=(Î»_{1},âŠ,Î»_{n}) of *n* if, in the decomposition of *Ï* as a product of disjoint cycles, there are Î»_{j} cycles of length *j*. Note that Î»(*Ï*) is a partition of *n* in the sense that Î»_{j}â„0 and . For example, Î»_{1} is the number of fixed points of *Ï* and Î»_{n}=1 if and only if *Ï* is an *n*-cycle.

For each partition Î»âą*n*, the probability that a random permutation on *n* letters has cycle structure *Ï* is given by Cauchy's formula [16, ch. 1]:
1.22

For of positive degree *n*, we say its cycle structure is Î»(â*f*)=(Î»_{1},âŠ,Î»_{n}) if, in the prime decomposition (we allow repetition), we have . Thus, we get a partition of *n*. In analogy with permutation, Î»_{1}(â*f*) is the number of roots of *f* in (with multiplicity) and *f* is irreducible if and only if Î»_{n}(â*f*)=1.

For a partition Î»âą*n*, we let *Ï*_{Î»} be the characteristic function of of cycle structure Î»:
1.23
The Prime Polynomial Theorem gives the mean values of *Ï*_{Î»}:
1.24
as (see lemma 2.1). We prove independence of cycle structure of shifted polynomials:

### Theorem 1.4

*For fixed positive integers n and s we have*
*uniformly for all h*_{1}*,âŠ,h*_{s} *distinct polynomials in* *of degrees* *and on all partitions Î»*_{1}*,âŠ,Î»*_{s}*âąn as* .

### Remark

In this theorem, Î»_{1},âŠ,Î»_{s} are partitions of *n* and are not the same as the Î»_{1},âŠ,Î»_{n} that appear in the definition of Î»(â*f*) or Î»(*Ï*) where in that case the Î»_{i} are the number of parts of length *i*.

We note that the statistic of theorem 1.4 is induced from the statistics of the cycle structure of tuples of elements in the direct product of *s* copies of the symmetric group on *n* letters *S*_{n}. This plays a role in the proof, where we use that a certain Galois group is [17], and we derive the statistic from an explicit Chebotarev theorem. Since we have not found the exact formulation that we need in the literature, we provide a proof in the appendix.

## 2. Mean values

For the reader's convenience, we prove in this section some results for which we did not find a good reference. We define the *norm* of a non-zero polynomial to be |*f*|=*q*^{deg(âf)} and set |0|=0.

We start by proving (1.24):

### Lemma 2.1

*If* Î»âą*n* *is a partition of* *n* *and* *n* *is a fixed number then*
2.1
as .

### Proof.

To see this, note that to get a monic polynomial with cycle structure Î», we pick any Î»_{1} primes of degree 1, Î»_{2} primes of degree 2 (irrespective of the choice of ordering), and multiply them together. Thus
2.2
where *Ï*_{A}(*j*) is the number of primes of degree *j* in . By the Prime Polynomial Theorem, *Ï*_{A}(*j*)=*q*^{j}/*j*+*O*(*q*^{j/2}/*j*) whenever *j*â„2 and *Ï*_{A}(1)=*q*. Hence *Ï*_{A}(*j*)=*q*^{j}/*j*+*O*(*q*^{jâ1}/*j*). So
2.3
which by (1.22) gives (2.1).ââȘ

Next, we prove (1.16):

### Lemma 2.2

*The mean value of* *d*_{k}(â*f*) *is*
2.4

### Proof.

The generating function for *d*_{k}(â*f*) is the *k*th power of the zeta function associated to the polynomial ring :
2.5
Here,
2.6
Using the Taylor expansion
2.7
and comparing the coefficients of *u*^{n} in (2.5) gives
2.8
as needed.ââȘ

## 3. Proof of theorem 1.4

In the course of the proof, we shall use the following explicit Chebotarev theorem, which is a special case of theorem A.4 of appendix A:

### Theorem 3.1

*Let* *A**=(A*_{1}*,âŠ,A*_{n}*) be an n-tuple of variables over* *let* *be monic, separable and of degree m viewed as a polynomial in t, let L be a splitting field of* *over* *and let* *. Assume that* *is algebraically closed in L. Then there exists a constant* *such that for every conjugacy class CâG we have*

Here Fr_{a} denotes the Frobenius conjugacy class ((*S*/*R*)/*Ï*) in *G* associated to the homomorphism given by , where and *S* is the integral closure of *R* in the splitting field of . See appendix A, in particular (A.51), for more details.

Let ** A**=(

*A*

_{1},âŠ,

*A*

_{n}) be an

*n*-tuple of variables and set 3.1 where the

*h*

_{i}are distinct polynomials. Let

*L*be the splitting field of over and let be an algebraic closure of . By [17, Proposition 3.1], In [17], it is assumed that

*q*is odd, but using [18] that restriction can now be removed for

*n*>2. This, in particular, implies that (since the image of the restriction map is , so by the above and Galois correspondence, , and in particular ). Hence, we may apply theorem 3.1 with the conjugacy class to get that Since |

*C*|/|

*G*|=

*p*(Î»

_{1})âŻ

*p*(Î»

_{s}) and since , it remains to show that for with we have Fr

_{a}=

*C*if and only if for all

*i*=1,âŠ,

*s*.

And indeed, extend the specialization ** A**âŠ

**to a homomorphism**

*a**ÎŠ*of to , where

**=(**

*Y**Y*

_{ij}), and

*Y*

_{i1},âŠ,

*Y*

_{in}are the roots of . Then Fr

_{a}is, by definition, the conjugacy class of the Frobenius element Fr

_{ÎŠ}â

*G*, which is defined by 3.2 Note that Fr

_{ÎŠ}permutes the roots of each and hence can be identified with an

*s*-tuple of permutations . Since the

*ÎŠ*(

*Y*

_{ij}) are distinct, the cycle structure of

*Ï*

_{i}equals the cycle structure of the

*ÎŠ*(

*Y*

_{ij})â

*ÎŠ*(

*Y*

_{ij})

^{q},

*j*=1,âŠ,

*n*by (3.2), which in turn equals the cycle structure of the polynomial . Hence

*Fr*

_{ÎŠ}â

*C*if and only if for all

*i*, as needed.ââȘ

## 4. Proof of theorem 1.1

First, we need the following lemma:

### Lemma 4.1

*Let* *and* *such that deg*(*h*)<*n*. *Then we have that*
4.1

### Proof.

The number of square-free is *q*^{n}â*q*^{nâ1} for *n*â„2 (for *n*=1 it is *q*), and since , as *f* runs over all monic polynomials of degree *n* so does *f*+*h*, and hence the number of such that *f*+*h* is square-free is also *q*^{n}â*q*^{nâ1}. Therefore, there are at most 2*q*^{nâ1} monic for which at least one of *f* and *f*+*h* is not square-free, as claimed.ââȘ

We denote by the mean value of an arithmetic function *A* over :
4.2

For this, it follows that if *A* is an arithmetic function on that is bounded independently of *q*, then
4.3

Now for square-free *f*, the divisor function *d*_{k}(â*f*) depends only on the cycle structure of *f*, namely
4.4
where for a partition Î»=(Î»_{1},âŠ,Î»_{n}) of *n*, we denote by the number of parts of Î». Therefore, we may apply (4.3) with (4.4) to get
4.5
Since the function *k*^{Î»(âf)} depends only on the cycle structure of *f*, it follows from theorem 1.4 that
4.6
Applying again (4.3) with (4.4) together with lemma 2.2, we conclude that
4.7
Combining (4.5), (4.6) and (4.7) then gives the desired result.ââȘ

## 5. Proof of theorem 1.2

We argue as in Â§4: (Here the first passage uses (4.3) with (4.4), the last also uses lemma 2.2, and the middle passage is done by invoking theorem 1.4.)

## 6. Proof of theorem 1.3

Let be the characteristic function of the primes of degree *n*, i.e.
6.1
The Prime Polynomial Theorem gives that and we have calculated in Â§4 that . Since these two functions clearly depend only on cycle structures (recall that *Î±*â 0), theorem 1.4 gives
6.2
Therefore,
as needed.

## 7. Comparing conjectures and our results

In this section, we check the compatibility of the theorems presented in Â§1c with the known results over the integers.

### (a) Estermann's theorem for

First, we prove the function field analogue of Estermann's result (1.2). For simplicity, we carry it out for *h*=1.

### Theorem 7.1

*Assume that nâ„1. Then*
7.1
*(Note that q is fixed in this theorem).*

We need two auxiliary lemmas before proving theorem 7.1.

Let be monic polynomials. We want to count the number of monic polynomial solutions of the linear Diophantine equation
7.2
As follows from the Euclidean algorithm, a necessary and sufficient condition for the equation *Au*â*Bv*=1 to be solvable in is gcd(*A*,*B*)=1.

### Lemma 7.2

*Given monic polynomials* *gcd*(*A*,*B*)=1 *and*
7.3
*then the set of monic solutions* (*u*,*v*) of *(7.2)* *forms a non-empty affine subspace of dimension* *n*âdeg(*A*)âdeg(*B*), *hence the number of solutions is exactly* *q*^{n}/|*A*||*B*|.

### Proof.

We first ignore the degree condition. By the theory of the linear Diophantine equation, given a particular solution , all other solutions in are of the form 7.4 where runs over all polynomials.

Given *u*_{0}, we may replace it by *u*_{1}=*u*_{0}+*kB* where deg(*u*_{1})<deg(*B*) (or is zero), so that we may assume that the particular solution satisfies
7.5
In that case, if *k*â 0 then
7.6
and *u*_{0}+*kB* is monic if and only if *k* is monic. Hence if *k*â 0, then
7.7

Thus, the set of solutions of (7.2) is in one-to-one correspondence with the space of monic *k* of degree *n*âdeg(*A*)âdeg(*B*). In particular, the number of solutions is *q*^{n}/|*A*||*B*|.ââȘ

Let 7.8 Then we have the following lemma.

### Lemma 7.3

*For* *Î±*+*ÎČ*=*n*=*Îł*+*ÎŽ*,
7.9

### Proof.

We have some obvious symmetries from the definition
7.10
and hence to evaluate *S*(*Î±*,*ÎČ*;*Îł*,*ÎŽ*) it suffices to assume
7.11
Assuming (7.11), we write
7.12
Note that *Î±*,*Îł*â€*n*/2 (since *Î±*+*ÎČ*=*n* and *Î±*â€*ÎČ*) and hence . Thus, we may use lemma 7.2 to deduce that
7.13
and therefore
7.14
Recall the MĂ¶bius inversion formula, which says that, for monic *f*, equals 1 if *f*=1, and 0 otherwise. Hence, we may write the coprimality condition using the MĂ¶bius function as
7.15
and therefore
7.16
where we have used the fact that *Î±*â€*ÎČ* and *Îł*â€*ÎŽ*.

We next claim that 7.17 which when we insert into (7.16) proves the lemma.

To prove (7.17), we sum over *d* of fixed degree
7.18
and recall that [19, ch. 2, exercise 12]
7.19
from which (7.17) follows.ââȘ

### Proof of theorem 7.1

We write 7.20 We partition this into a sum over variables with fixed degree, that is 7.21

We now input the results of lemma 7.3 into (7.21) to deduce that
7.22
Of the (*n*+1)^{2} quadruples of non-negative integers (*Î±*,*ÎČ*;*Îł*,*ÎŽ*) so that *Î±*+*ÎČ*=*n*=*Îł*+*ÎŽ*, there are exactly 4*n* tuples (*Î±*,*ÎČ*;*Îł*,*ÎŽ*) for which , namely they are
7.23
and the 4(*n*â1) tuples of the form
7.24
for 0<*i*<*n*.

Concluding, we have 7.25 proving the theorem.ââȘ

It is easy to check that theorem 1.1 is compatible with the function field analogue of Estermann's result. Taking in (7.1), we recover the same results as presented in (1.17) with *k*=2.

### (b) Higher divisor functions

Next, we want to check compatibility of our result in theorem 1.1 with what is conjectured over the integers. It is conjectured that
7.26
where *P*_{2(kâ1)}(*u*;*h*) is a polynomial in *u* of degree 2(*k*â1), whose coefficients depend on *h* (and *k*). This conjecture appears in the work of IviÄ [20] and Conrey & Gonek [5], and from their work, with some effort, we can explicitly write the conjectural leading coefficient for the desired polynomial. The conjecture over states that
7.27
where
7.28
with
7.29
where *e*(*x*)=e^{2Ïix} and *c*_{m}(*h*) is the Ramanujan sum,
7.30

We now translate the conjecture above to the function field setting using the correspondence and and that summing over positive integers correspond to summing over monic polynomials in . Under this correspondence, the function field analogue of the above polynomial is given in the following conjecture.

### Conjecture 7.4

*For* *q* *fixed, let* . *Then as* ,
7.31
*where*
7.32
*where* |*m*|=*q*^{deg(m)},
7.33
*and*
7.34
*is the Ramanujan sum over* . *The sum above is over all monic polynomials* , *ÎŒ*(â*f*) *is the MĂ¶bius function for* *and* *ÎŠ*(*m*) *is the* *analogue for Euler's totient function*.

### Remark 7.5

Note that 7.35 corresponds to as given in (7.29).

### Remark 7.6

Note that we establish this conjecture for *k*=2 and *h*=1 in theorem 7.1.

We now check that our theorem 1.1 is consistent with the conjecture (7.27) and (7.32) for the leading term of the polynomial *P*_{2(kâ1)}(*u*;*h*).

The polynomial given by theorem 1.1 is 7.36

We wish to show that, as , *A*_{k,q}(*h*)/[(*k*â1)!]^{2} matches the leading coefficient of , that is
7.37

Indeed, from (7.34) we note that |*c*_{m,q}(*h*)|=*O*_{h}(1), and it is easy to see that
7.38
Thus, we find
7.39
The series in the *O* term is a geometric series:
7.40
and hence tends to 0 as , giving (7.37).

## Funding statement

J.C.A. is supported by an IHĂS Postdoctoral Fellowship and an EPSRC William Hodge Fellowship. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 320755 and from the Israel Science Foundation (grant no. 925/14).

## Acknowledgements

We thank an anonymous referee for detailed comments and suggestions.

## Appendix A. An explicit Chebotarev theorem

We prove an explicit Chebotarev theorem for function fields over finite fields. This theorem is known to experts, cf. [21, Theorem 4.1], [22, Proposition 6.4.8] or [23, Theorem 9.7.10]. However, there it is not given explicitly with the uniformity that we need to use. Therefore, we provide a complete proof.

**(a) Frobenius elements**

Let be a finite field with *q* elements and algebraic closure . We denote by Fr_{q} the Frobenius automorphism *x*âŠ*x*^{q}.

Let *R* be an integrally closed finitely generated -algebra with fraction field *K*, and let be a monic separable polynomial of degree such that
A 1
is invertible. Let ** Y**=(

*Y*

_{1},âŠ,

*Y*

_{m}) be the roots of , and put We identify

*G*with a subgroup of

*S*

_{m}via the action on

*Y*

_{1},âŠ,

*Y*

_{m}: A 2 By (A 1) and Cramer's rule,

*S*is the integral closure of

*R*in

*L*and

*S*/

*R*is unramified. In particular, the relative algebraic closure of in

*L*is contained in

*S*. For each

*Îœ*â„0 we let A 3 the preimage of in

*G*under the restriction map. Since is commutative,

*G*

_{Îœ}is stable under conjugation.

For every with there exists a unique element in *G*, which we call the *Frobenius element* and denote by
A 4
such that
A 5
Since *S* is generated by ** Y** over

*R*, it suffices to consider

*x*â{

*Y*

_{1},âŠ,

*Y*

_{k}} in (A 5). If we further assume that , then (A 5) gives that [

*S*/

*R*/

*ÎŠ*]

*x*=

*x*

^{qÎœ}for all , hence A 6

### Lemma A.1

*For every* *g*â*S*_{m} *and* *Îœ*â„1 *there exists* *such that* Fr_{qÎœ} *acts on the rows of* *V* _{g,Îœ} *as* *g* *acts on* ** Y**:
A 7

### Proof.

By replacing *q* by *q*^{Îœ}, we may assume without loss of generality that *Îœ*=1. By relabelling, we may assume without loss of generality that
A 8
where *s*_{1}=1, *s*_{i+1}=*e*_{i}+1 and *e*_{k}=*m*.

Let *V* be the block diagonal matrix
where
is the Vandermonde matrix corresponding to an element of degree Î»_{i}=*e*_{i}â*s*_{i} over . So , hence *V* is invertible, and by definition Fr_{q} acts on the rows of *V* as the permutation *g*.ââȘ

### Lemma A.2

*Let* *with* *and let* *g*â*G*_{Îœ}. *Then*
A 9
*where* *V* =*V* _{g,Îœ} *is the matrix from lemma A.1*.

### Proof.

Let be the unique solution of the linear system
A 10
i.e.
If , i.e. , we get by applying Fr_{qÎœ} on (A 10) that
Hence [(*S*/*R*)/*ÎŠ*]=*g* by (A 5).

Conversely, if [(*S*/*R*)/*ÎŠ*]=*g*, then *ÎŠ*(*Y* _{i})^{qÎœ}=*ÎŠ*(*Y* _{g(i)}) by (A 2) and (A 5). We thus get that Fr_{qÎœ} permutes the equations in (A 10), hence Fr_{qÎœ} fixes the unique solution of (A 10). That is to say, , as needed.ââȘ

Next, we describe the dependence of the Frobenius element when varying the homomorphisms. For we define
A 11
Unlike the case when working with ideals, this set is not a conjugacy class in *G*, as we fix the action on . However, as we will prove below, the group *G*_{0} acts regularly on ((*S*/*R*)/*Ï*) by conjugation. In particular, if *G*_{0}=*G*, or equivalently if (with denoting an algebraic closure of ), then ((*S*/*R*)/*Ï*) is a conjugacy class.

To state the result formally, we recall that a group *Î* acts *regularly* on a set *Î©* if the action is free and transitive, i.e. for every *Ï*_{1},*Ï*_{2}â*Î©* there exists a unique *Îł*â*Î* with *ÎłÏ*_{1}=*Ï*_{2}.

### Lemma A.3

*Let* *and let* *H* *be the subset of* *consisting of all homomorphisms prolonging* *Ï*. *Assume that* .

(1)â

*The group**G*_{0}*defined in*(A 3)*acts regularly on**H**by**g*:*ÎŠ*âŠ*ÎŠ*Â°*g*.(2)â

*For every**g*â*G*_{0}*and**ÎŠ*â*H*,*we have*(3)â

*Let**ÎŠ*â*H*,*let**g*=[*S*/*R*/*ÎŠ*],*let**H*_{g}={*Îš*â*H*:[*S*/*R*/*Îš*]=*g*}*and let**C*_{G0}(*g*)*be the centralizer of**g**in**G*_{0}.*Then**C*_{G0}(*g*)*acts regularly on**H*_{g}.(4)â#

*H*_{g}=#*G*_{0}/#*C*=#*G*/*ÎŒ*â #*C*,*where**C**is the conjugacy class of**g*in*G*_{0}.

### Proof.

We consider *G*_{0}â€*G* as subgroups of *S*_{m} via the action on *Y* _{1},âŠ,*Y* _{m}. Let *g*â*G*_{0} and *ÎŠ*â*H*. Then *g*(*x*)=*x* and *ÎŠ*(*x*)=*x*, thus *ÎŠ*Â°*g*(*x*)=*x*, for all . Thus, *ÎŠ*Â°*g*â*H*. If *ÎŠ*Â°*g*=*ÎŠ*, then *ÎŠ*(*Y* _{g(i)})=*ÎŠ*(*Y* _{i}) for all *i*. Since it follows that , thus *ÎŠ* maps {*Y* _{1},âŠ,*Y* _{m}} injectively onto {*ÎŠ*(*Y* _{1}),âŠ,*ÎŠ*(*Y* _{m})}. We thus get that *Y* _{g(i)}=*Y* _{i}, hence *g* is trivial. This proves that the action is free.

Next, we prove that the action is transitive. Let *ÎŠ*,*Îš*â*H*. Then ker*ÎŠ* and ker*Îš* are prime ideals of *S* that lie over the prime ideal ker*Ï* of *R*, hence over the prime of . By [24, VII, 2.1], there exists such that . Replace *ÎŠ* by to assume without loss of generality that ker*ÎŠ*=*kerÎš*. Hence *ÎŠ*=*Î±*Â°*Îš*, where *Î±* is an automorphism of the image *ÎŠ*(*S*)=*Îš*(*S*) that fixes both and . That is to say, , where *Ï* is a common multiple of *Îœ* and *ÎŒ*. By (A 5)
so *ÎŠ*=*Îš*Â°*g*, where *g*=[(*S*/*R*)/*Îš*]^{Ï/Îœ}. Since, for we have *g*(*x*)=*x*^{qÏ} and *ÎŒ*|*Ï*, we have *g*(*x*)=*x*, so *g*â*G*_{0}. This finishes the proof of (1).

To see (2) note that
so *g*[(*S*/*R*)/*ÎŠ*Â°*g*]=[(*S*/*R*)/*ÎŠ*]*g* (since *ÎŠ* is unramified), as claimed.

The rest of the proof is immediate, as (3) follows immediately from (1) and (2), and (4) follows from (3).ââȘ

By (A 6) and lemma A.3, it follows that if , then ((*S*/*R*)/*Ï*)â*G*_{Îœ} is an orbit of the action of conjugation from *G*_{0}.

Let *C*â*G* be such an orbit, i.e. *C*=*C*_{g}={*hgh*^{â1}:*h*â*G*_{0}}, *g*â*G*_{Îœ}. Then *C*â*G*_{Îœ}, since the latter is stable under conjugation (see after (A 3)). The explicit Chebotarev theorem gives the asymptotic probability that ((*S*/*R*)/*Ï*)=*C*:

### Theorem A.4

*Let Îœâ„1, let CâG*_{Îœ} *be an orbit of the action of conjugation from G*_{0}*. Then*
*as* .

We define cmp(*R*) below.

Before proving this theorem, we need to recall the LangâWeil estimates, which play a crucial role in the proof of the theorem and in particular give the asymptotic value of the denominator of *P*_{Îœ,C}.

Let *U* be a closed subvariety of that is geometrically irreducible. LangâWeil estimates give that
A 12
Note that both *n* and are stable under base change. This may be reformulated in terms of -algebras, to say that if
A 13
then
A 14
provided is a domain, where *cmp*(*R*) is a function of and *n*, taking minimum over all presentations (A 13). By the remark following (A 12), it follows that if two -algebras *S* and *S*âČ become isomorphic over , then cmp(*S*âČ) is bounded in terms of cmp(*S*). A final property needed is that if *R*â*S* is a finite map of degree *d*, then cmp(*S*) is bounded in terms of cmp(*R*) and *d*.

### Proof.

Let *g*â*C*, let *V* =*V* _{g,Îœ} be as in (A 7) and let *S*âČ=*R*[** Z**], where

**=**

*Z**V*

^{â1}

**. Note that**

*Y***is the unique solution of the linear system A 15**

*Z*Let . By (A 9), the number of with [(*S*/*R*)/*ÎŠ*]=*g* equals *N*. By lemma A.3, for each *Ï* there exist exactly #*G*_{0}/#*C* homomorphisms with [(*S*/*R*)/*ÎŠ*]=*g* prolonging *Ï*. Hence,
Since *G*_{Îœ} is a coset of *G*_{0}, #*G*_{0}=#*G*_{Îœ}. Hence, it suffices to prove that . As *R*â*S*âČ is a finite map of degree , we get that and cmp(*S*âČ) is bounded in terms of *cmp*(*R*) and . It suffices to show that since then by (A 14) we have
and the proof is done.

Let *L* be the fraction field of *S* and *K* of *R*. Since *L*/*K* is Galois and and since the actions of Fr_{qÎœ} and *g* agree on , it follows that there exists an automorphism *Ï* of such that *Ï*|_{L}=*g* and . By (A 7) *Ï* permutes the equations (A 15), hence fixes ** Z** and thus

*S*âČ. In particular, if , then

*x*

^{qÎœ}=

*Ï*(

*x*)=

*x*, so , as was needed to complete the proof.ââȘ

## Footnotes

One contribution of 8 to a Theo Murphy meeting issue âNumber fields and function fields: coalescences, contrasts and emerging applicationsâ.

- Â© 2015 The Author(s) Published by the Royal Society. All rights reserved.