Correlations between primes in short intervals on curves over finite fields
Efrat Bank Tyler Foster








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Efrat Bank University of Michigan Mathematics Department 530 Church Street Ann Arbor, MI 48109-1043 United States
mailto:ebank@umich.eduebank@umich.edu


Tyler Foster L'Institut des Hautes Études Scientifiques Le Bois-Marie, 35 route de Chartres 91440 Bures-sur-Yvette France
mailto:foster@ihes.frfoster@ihes.fr



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Correlations between primes in short intervals
 on curves over finite fields
    Tyler Foster
    December 30, 2023
============================================================================





-.5cm
	We prove an analogue of the Hardy-Littlewood conjecture on the asymptotic distribution of prime constellations in the setting of short intervals in function fields of smooth projective curves over finite fields.








Introduction

	In recent work <cit.>, Bary-Soroker and the first author use their work with Rosenzweig <cit.> on the asymptotic distribution of primes inside short intervals in _q[t] to establish a natural counterpart, over the field _q(t), to the still unsolved Hardy-Littlewood conjecture. In <cit.>, the two present authors show how to extend the results of <cit.> to give asymptotic distributions of primes in short intervals on the complement of a very ample divisor in a smooth, geometrically irreducible projective curve C over a finite field _q. In the present paper, we apply ideas from <cit.> and <cit.> to prove a natural counterpart to the Hardy-Littlewood conjecture on the complement of a very ample divisor in a smooth, geometrically irreducible projective curve C over a finite field _q.
	
The Hardy-Littlewood conjectures for short intervals
	Let K be a number field, 𝒪_K its ring of integers, and N(h)#𝒪_K/(h) the norm on elements h∈𝒪_K. Given an n-tuple σ=(σ_1,...,σ_n) of elements in 𝒪_K, denote by π_K,σ(x) the n-tuple principal prime counting function
	
    π_K,σ(x)
    	  #{ h∈𝒪_K : 2<N(h)≤ x, and each (h+σ_i)⊂𝒪_K is prime}.

The Hardy-Littlewood n-tuple conjecture <cit.> (henceforth the HL conjecture), asserts that:
	

The function π_,σ satisfies the asymptotic formula
	
    π_,σ(x) ∼ S(σ) x/(log x)^n,     x→∞

for a positive constant S(σ).


	Despite much numerical verification and many proof attempts, the HL conjecture remains open. Results toward the conjecture include the work of Goldston, Pintz and Yildirim <cit.>, Zhang <cit.> and Maynard <cit.>. See Graville <cit.> for a partial history of the problem.

	One can also formulate short interval variants of the HL conjecture. As in <cit.>, the ambiguity in defining subsets of 𝒪_K in terms of the norm on K leads to at least two possible formulations. Fix a n-tuple σ=(σ_1,...,σ_n) of elements in 𝒪_K. Then each real number 1>ε>0 determines a family of intervals I(x,ε) [x-x^ε, x+x^ε]⊂, with corresponding prime counting function
	
    π_K,σ(I(x,ε))
    	  #{ N_K(h)∈ I(x,ε):each (h+σ_i)⊂𝒪_K is prime}.

On the other hand, if S={K} and ε_S=(ε_𝔭)_𝔭∈ S is an #S-tuple of real numbers 1>ε_𝔭>0, then for each b∈𝒪_K, we can define
	
    I(b,ε_S){ a∈𝒪_K :  |a-b|_𝔭≤ |b|^ε_𝔭_𝔭𝔭∈ S },

with corresponding prime counting function
	
    π_K,σ(I(b,ε_S))
    	 = #{a∈ I(b,ε_S):(a+σ_i)⊂𝒪_K}.
	
 

(i). The function π_K,σ(I(x,ϵ)) satisfies
	
    π_K,σ(I(x,ϵ))
    	 ∼ S(σ)#I(x,ϵ)/(log x)^n,     x→∞,

where S(σ) is some positive constant depending on σ and on the class number of K.

	(ii). The function π_K,σ(I(b,ε_S)) satisfies
	
    pi_K,σ(I(b,ε_S))
    	  ∼  S(σ)#I(b,ε_S)/(log N_K(b))^n,
    	     N_K(b)→∞,

where S(σ) is some positive constant depending on σ and on the class number of K.	
	 

Gross and Smith <cit.> present numeric evidence for Conjecture <ref>.(i) for certain number fields and reasonable sets, which are regions that may be interpreted as short intervals. Based on this, they present a conjecture similar to Conjecture <ref>.(i) above.
	
	Notice that when K=, then S={∞} and Conjectures <ref>.(i) and (ii) coincide. Natural analogues of Conjecture <ref> and <ref> also hold when we replace  with field of rational functions _q(t) over a finite field _q in the large q limit. For f∈_q[t] monic and ε>0 a real number, define
	I(f,ε)
	{h∈_q[t] : | h-f| ≤ | f |^ε},
with |f| q^ f. For n-tuples 𝐟=(f_1,...,f_n) of distinct polynomials f_i and distinct polynomials in _q[t], define
	
    π_q,𝐟(k)
    	  #{ h∈_q[t]: h monic of degree  k, and each  h+f_i is a prime polynomial};

	
    [                                    π_q,𝐟(I(f,ε))                                                  #{h∈ I(f,ε): each  h+f_i is a prime polynomial}. ]

	

	Fix an integer B>n. Then:
	
	(i) (Pollack <cit.>, Bary-Soroker <cit.>, Carmon <cit.>).
	The function π_q,f(k) satisfies
	
    π_q,𝐟(k) = q^k/k^n(1 + O_B(q^-1/2))

uniformly in f_1,...,f_n with degrees bounded by B as q∞.
.2cm
	
	(ii)
(Bank & Bary-Soroker <cit.>). The function π_q,𝐟(I(f_0,ε)) satisfies
	
	
    π_q,𝐟(I(f_0,ε))
    	 = # I(f_0,ε)/i=1n(f_0+f_i)( 1 + O_B(q^-1/2))

uniformly for all f_1,…,f_n of degree at most B, and for all monic polynomials f_0 satisfying 2/ε< f_0<B, as q∞ an odd prime power.
	
	



	
Main results: Correlation of primes in short intervals on curves 
		Let C be a smooth projective geometrically irreducible curve over a finite field _q. Fix an effective divisor E=m_1𝔭_1+⋯ +m_s𝔭_s on C. Let σ  (σ_1,...,σ_n) be an n-tuple of rational functions σ_i on C, regular on C\ E C\supp(E). Define an n-tuple principal prime counting function 
	
    π_C,σ(E)
    	    #{[                               h∈ K(C)div(h)_-=E,;                         function h+σ_1, …, h+σ_n; ideal in the ring of regular functions on   C\ E ]}.


	The condition that div(h)_-=E should be thought of as analogous to the condition h=k in (<ref>). In the special case where C=^1, the ring of rational functions on C regular away from the point ∞∈^1 is the polynomial ring _q[t]. Thus when E=∞, the function π_C,σ defined in (<ref>) reduces to the function π_q,f defined in (<ref>).



	The function π_C,σ(E) satisfies
	
    π_C,σ(E) ∼ S(σ) q^deg_ E/(deg_ E)^n,     q →∞

for a positive constant S(σ) depending on σ, uniformly in σ_1,...,σ_n with bounded degree.



Given a regular function f on C\ E, the interval (of size E around f) is the set
	
    [                                                     I(f,E)                                                            {[                 C\ E such; ν_𝔭_i(h-f)≥ -m_i  1≤ i≤ s ]};                                                                                                                     =                                             f+H^0(C,𝒪(E)). ]

The interval I(f,E) is a short interval if the order of the pole of f at each 𝔭_i is at least m_i, and strictly greater than m_i for at least one 𝔭_i. Define
	
    π_C,σ(I(f,E))
    	    #{[ h∈ I(f,E) such that h+σ_1,...,h+σ_n generate prime;   ideals in the ring of regular functions on  C\ E ]}.



	Our main result is an analogue of Conjecture <ref> that extends Theorem <ref>.(ii) to curves of arbitrary genus over _q. Fix a smooth projective geometrically irreducible curve C over _q. Let E=m_1𝔭_1+⋯+m_r𝔭_r be an effective divisor on C, and let f_0,σ_1,...,σ_n be distinct regular functions on C\ E satisfying -ν_𝔭(f_0)> m_𝔭 and ν_𝔭(f_0) ≠ν_𝔭(σ_i) for each 1≤ i≤ n.


	Fix an integer B>n. If char_ _q2 and E≥ 3E_0 for some effective divisor E_0 on C with deg_ E_0≥2g+1, then the asymptotic formula
	
    π_C,σ(I(f_0,E))
    	 = #I(f_0,E)/∏_i=1^n (f_0+σ_i)(1+O_B(q^-1/2))

holds uniformly for all E and f_0,σ_1,…,σ_n as above satisfying deg(div(f_0+σ_i)|_E)<B, and as q∞ an odd prime power.





	
Outline of paper
	Our strategy in proving Theorem <ref> is similar in spirit to <cit.>. In <ref> we briefly review the necessary divisor theory and show that the splitting fields of distinct linear functions, evaluated at the generic element ℱ_A of a short interval, are linearly disjoint. 
We use this to show that the Galois group of the product of several linear functions evaluated at ℱ_A is isomorphic to a direct product of symmetric groups. In <ref> we use a Chebotarev-type density theorem to estimate π_C,σ(I(f_0,E)). In <ref>, we prove the main Theorem <ref> as a special case of the more general Theorem <ref> , which deals with general factorization types.




	
Acknowledgments
	The authors would like to thank Lior Bary-Soroker for suggesting a version of this problem. We would like to thank Jeff Lagarias and Mike Zieve for many helpful conversations. We also extend a warm thank you to Jordan Ellenberg, Alexei Entin, and Zeev Rudnick for helpful perspectives. 
	
	 The research that lead to this paper was conducted while the first author was at the University of Michigan and while  the  second  author  was  a  visiting  researcher  at  L`Institut  des  Hautes Études Scientifiques and at L'Institut Henri Poincaré. The first author thanks the AMS-Simons Travel Grant for supporting her visit to IHES. The second author thanks Le Laboratoire d`Excellence CARMIN for their financial support.


	





Galois group calculation

Relevant background
	We make use of the theory of divisors on algebraic varieties. Necessary background appears in <cit.>, with further details in <cit.>. For each point 𝔭∈ C, let ν_𝔭 denote valuation at 𝔭, applied to both functions and divisors on C.
	
	For the remainder of the present <ref>, we fix an effective very ample divisor E on C and a function f_0 regular on C\ E satisfying
	
    -ν_𝔭(f_0)
    	 > ν_𝔭(E)
    	         𝔭∈supp(E).

We require a somewhat stronger positivity condition on E than “effective and very ample." Namely, assume that there exists a divisor E_0 on C such that
	
    E ≥ 3E_0.

	
	Observe however that if E is effective and very ample but fails to satisfy (<ref>), then we can replace E by 3E to achieve (<ref>).
Let σ=(σ_1,...,σ_n) be an n-tuple of distinct rational functions on C, each regular on C\ E, and each satisfying the inequalities
	
    ν_𝔭(σ_i)
    	 ≠ ν_𝔭(f_0)
    	      
    	-ν_𝔭(σ_i)
    	 > ν_𝔭(E),
    	         𝔭∈supp(E).

For each σ_i, define the monic linear polynomial 
	
    L_i(X)  σ_i+X
    	      in      _q(C)[X].

	
	Let R denote the ring of regular functions on C\ E. Fix a basis {1,g_1,...,g_m} of H^0(C,𝒪(E)) once and for all, let _q(A)=_q(A_0, . . . , A_m) denote the field of rational functions in m+1 variables, and define 𝔸^m+1Spec_ _q[A]=Spec_ _q[A_0,...,A_m]. On the trivial family of curves (C\ E)×__q𝔸^m+1=Spec_ R[A]=Spec_ R⊗__q_q[A], we have the regular function
	
    ℱ_𝐀  
    	f_0+A_0+∑_j=1^mA_j g_j.

For each _q-rational point a∈𝔸^m+1, the restriction of ℱ_A to (C\ E)×__qSpec_ κ(a) defines a regular function ℱ_a on C\ E.

.2cm

	Let K/_q(𝐀) be an algebraic extension. For an ideal ℑ in the Dedekind domain K⊗__q(𝐀)R(𝐀), let
	
    ℑ = 𝔓^e_1_1⋯𝔓^e_ℓ_ℓ

denote the prime decomposition of ℑ.


(i) If e_1=⋯=e_ℓ=1 in (<ref>), then we define the splitting field of ℑ over K, denoted split(ℑ) or split(ℑ/K), to be the composite
	
    split(ℑ)
    	  split(𝔓_1)⋯split(𝔓_ℓ),

where for each 1≤ i≤ n, split(𝔓_i) denotes the normal closure of κ(𝔓_i) in _q(A).
	
	(ii)
	If each extension κ(𝔓_i)/K is separable, then the composite extension split(ℑ)/K is normal, and we define the Galois group of ℑ to be
	
    Gal(ℑ/K)
    	  Gal(split(ℑ)/K).





Linearly disjointness of splitting fields	
	For each 1≤ i≤ n, the rational function L_i(ℱ_A) on C__q(A) determines a morphism
	
    L_i(ℱ_A):C__q(A)⟶^1__q(A)

and a field extension
	
    _q(A) ↪ _q(C)(A_0,…,A_m).

Likewise, if we define _q(A')_q(A_1,...,A_m), then the rational function 	
	
    Ψ_i
    	  σ_i+f_0+∑_j=1^mA_j g_j
    	 = 
    	L_i(ℱ_A)-A_0.

on C__q(A') provides a morphism	
	
    Ψ_i:C__q(A')⟶^1__q(A')

and field extension
	
    _q(A')(t) ↪ _q(C)(A_1,…,A_m).



The field extension (<ref>) is separable.


	The assumption (<ref>) implies that for each 𝔭∈supp(E), we have
	
    ν_𝔭(σ_i +f_0)
    	 = min{ ν_𝔭(σ_i), ν_𝔭(f_0) } < 
    	-ν_𝔭(E).

This makes I(σ_i+f_0,E) a short interval in the sense of <cit.>. Thus by <cit.>, the variety V(L_i(ℱ_A))⊂ (C\ E)__q(A) consists of a single point 𝔓_i such that the field extension
	
    _q(A) ↪ κ(𝔓_i)

is separable. Under the identification
	
    [     _q(A)           _q(A')(t);       A_0         ⟼       -t, ]

the extensions (<ref>) and (<ref>) become isomorphic. Thus (<ref>) is separable.

   

	Let 𝔓_i be the underlying point of V(ℱ_A) as in the proof of Lemma <ref>. By Lemma <ref>, we can consider the splitting field
	
    split(L_i(ℱ_A))
    	  split(κ(𝔓_i)/_q(A))

obtained as the normal closure of the extension (<ref>), as defined in <cit.>. Let 𝔡_i denote the discriminant of the extension split(L_i(ℱ_A))/_q(A), as defined in <cit.> for instance. The discriminant is a fractional ideal in _q(A) that restricts to an actual ideal in _q(A')[A_0]. Because _q(A')[A_0] is a principal ideal domain, there exists a function D_i∈_q(A')[A_0], well defined up to multiplication by elements in _q(A')^×, such that
	
    𝔡_i = (D_i)
    	      _q(A')[A_0].

	
	Via the identification (<ref>), <cit.> implies that the map Ψ_i is ramified at a point x in C__q(A')\ E__q(A') if and only if D_i vanishes at the point Ψ_i(x) inside Spec_ _q(A')[A_0]⊂^1__q(A').


	Let Ω^1_C denote the sheaf of Kähler differentials on C. For each 1≤ i≤ n, the functions Ψ_i and L_i(ℱ_A) are regular on (C\ E)__q(A') and (C\ E)__q(A), respectively. Let dΨ_i and dL_i(ℱ_A) denote their differentials, and note that
	
    dΨ_i = dL_i(ℱ_A)
    	          C__q(A)\ E__q(A).

	
	Given a field extension K/_q, a section ω∈Γ(C_K\ E_K, Ω^1_C_K), and any point x∈ C_K\ E_K, let ω|_x denote the restriction of ω to the fiber (Ω^1_C_K)_x.

	



In the notation of Remark <ref>, D_1 and D_2 are relatively prime in the polynomial ring _q(A')[A_0] if and only if the system of equations
	 
    [ dΨ_1|_x       =      0;; dΨ_2|_y       =      0;;  Ψ_1(x)       = Ψ_2(y), ]

has no solution in pairs of (not necessarily distinct) points x,y∈(C\ E)__q(A').


	Because Ψ_i^-1(∞)=supp(E__q(A')), Remark <ref> implies that D_1 and D_2 are relatively prime if and only if the branch divisors of Ψ_1 and Ψ_2 are disjoint in Spec_ _q(A')[A_0]. Recall that the support of the branch divisor of Ψ_i is the set of all 𝔭∈^1__q(A') such that
	
    𝔭 = Ψ_i(x)
    	            
    	dΨ_i|_x = 0

for some x∈ C__q(A'). Thus a point 𝔭 of Spec_ _q(A')[A_0] in the support of both branch divisors Ψ_1 and Ψ_2 is any point satisfying Ψ_1(x)=𝔭=Ψ_2(y) and dΨ_1|_x=0=dΨ_2|_y for some (not necessarily distinct) pair of points x,y∈ C__q(A')\ E__q(A').



If char_ _q 2, then the system of equations (<ref>) has no solution in points x,y∈ C__q(A')\ E__q(A'). 


	Since Ψ_i=L_i(ℱ_A)-A_0, it suffices to prove that the system of equations
	 
    [ dL_1(ℱ_A)|_x            =            0; dL_2(ℱ_A)|_y            =            0;  L_1(ℱ_A)(x)            =  L_2(ℱ_A)(y) ]

has no solutions over _q(A).
	
	As in the proof of <cit.>, choose an effective very ample divisor E_0 satisfying (<ref>), define m_0_def=dim_ H^0(C,𝒪(E_0))-1, and let C⊂^m_0 be the closed embedding determined by E_0.

	Assume first that x≠ y. For each pair of distinct points ξ,η∈ C__q\ E__q, let t be a linear form on ^m_0__q such that t(ξ) t(η). The assumption on E_0 gives us a new _q-linear basis
	
    {1,t,t^2,…,t^ℓ,g_ℓ+1,…,g_m}      
    	H^0(C__q,𝒪(E__q)),

and a new coordinate system A_0,B_1,…,B_m on 𝔸^m+1__q such that
	
    L_i(ℱ_A)
    	 = σ_i+f_0+A_0+B_1t+⋯+B_ℓt^ℓ+B_ℓ+1g_ℓ+1+⋯+B_mg_m

for each i=1,2. For i=1,2, define Φ_i L_i(ℱ_A)-B_1t-B_2t^2.
	As in the proof of <cit.>, we can choose a Zariski opens U_ξη that provide a covering of
	
    ((C__q\ E__q)_q×(C__q\ E__q))\{diagonal},

such that an _q(A)-valued solution (u,v)∈ U_ξη to (<ref>) is the same thing as a solution to the single equation
	
    det(u,v)
    	    det(
    	[             1         2t(u)        φ_1(u);             1         2t(v)        φ_2(v);     t(v)-t(u) t(v)^2-t(u)^2        c(u,v) ])
    	  =  
    	0,

where φ_i-dΦ_i/dt for i=1,2, and where c(u,v)Φ_1(u)-Φ_2(v). A direct calculation gives
	
    det(u,v)
    	 = (t(v)-t(u)) (2 c(u,v)+(t(u)-t(v))(φ_1(u)+φ_2(v))).

By the same reasoning as in <cit.>, it suffices to prove that det(u,v) cannot be 0 identically on U_ξη.
	
	If n ≥ 3, then the coefficient of B_3 in 
	
    2 c(u,v)+(t(u)-t(v))(φ_1(u)+φ_2(v))

 is 2(t(u)^3-t(v)^3)+(t(u)-t(v)) 3(t(u)^2+t(v)^2), which is not identically 0.
Since t(u) t(v), this implies that det(u,v) is not identically zero on U_ξη.

	Assume next that x=y. If the pair x=y∈ C__q(A)\ E__q(A) is a solution to (<ref>), then L_1(ℱ_A)(x)=
L_2(ℱ_A)(x), implying that (σ_1-σ_2)(x)=0. Because σ_1 and σ_2 are distinct regular functions on C__q\ E__q, this implies that x∈ C__q\ E__q. 
This allows us to choose a linear form t on ^m_0__q such that t(x)≠ 0. Using this linear form t, choose an _q-linear basis (<ref>). This choice lets us write L_i(ℱ_A) as in (<ref>). The solution x satisfies dL_1(ℱ_A)|_x=0. But our choice of t lets us write
	
    0
    	 = 
    	dL_1(ℱ_A)|_x = dσ_1/dt(x)+df_0/dt(x)+B_1+∑_j=2^ℓj B_j t^j -1(x) + ∑_j=ℓ+1^mB_jdg_j/dt(x)

and so
	
    -B_1
    	 = dσ_1/dt(x)+df_0/dt(x)+∑_j=2^ℓj B_j t^j -1(x) + ∑_j=ℓ+1^mB_jdg_j/dt(x).

However, the right hand side of this last equation does not involve B_1, contradicting the fact that x∈ C__q\ E__q.

	
 
If char_ _q 2, then the splitting fields split(L_i(ℱ_𝐀)), for 1≤ i≤ n, are linearly disjoint over _q(A).


	As noted in the proof of Lemma <ref>, for each 𝔭∈supp(E) and for each 1≤ i≤ n, the conditions (<ref>) imply that the inequality (<ref>) holds. Thus the hypotheses of <cit.> are satisfied and we have (L_i(ℱ_𝐀),_q(A)) S_k_i with
	
    k_i  _ div(f_0+σ_i)|_C\ E = _ div(L_i(f_0))|_C\ E.

Because we have inclusions (L_i(ℱ_𝐀),_q(A))⊆(L_i(ℱ_𝐀),_q(A))↪ S_k_i, this implies
	
    (L_i(ℱ_𝐀),_q(A))
    	  
    	S_k_i.
 

	Let A_k_i⊂S_k_i denote the alternating group on k_i letters. Its fixed field is the quadratic extension
	
    split(L_i(ℱ_𝐀))^A_k_i ≅ _q(A)(√(D_i)),

where D_i∈_q(A')[A_0] is an element generating the discriminant ideal 𝔡_i as in Remark <ref>.
By <cit.>, it suffices to prove that the fields _q(A)(√(D_i)), for 1≤ i≤ n, are linearly disjoint.

	Without loss of generality, consider i=1,2. By Lemma <ref> and Proposition <ref>, the elements D_1,D_2∈_q(A')[A_0] have no common prime factors. Likewise, <cit.> implies that both D_1 and D_2 are square free. Because the fields _q(A)(√(D_1)) and _q(A)(√(D_2)) are degree-2 extensions of _q(A), their intersection _q(A)(√(D_1))∩_q(A)(√(D_2)) is either _q(A) itself, or else the two field extensions coincide: _q(A)(√(D_1))=_q(A)(√(D_2)). The latter is the case if and only if the product D_1D_2∈_q(A')[A_0] contains the square of a prime factor, contradicting the fact that D_1 and D_2 are square free and relatively prime in _q(A')[A_0].


Combining the results above, we have the following:
 For each algebraic extension K/_q, we have natural group isomorphisms


Gal( ni=1L_i(ℱ_𝐀), K(A) )
	  ≅  ni=1 Gal( L_i(ℱ_𝐀), K(A) )
	  ≅  
	S_k_1×⋯× S_k_n.
□



	







Counting argument and proof of Theorem <ref>
	In this section we prove Theorem <ref> the counting Proposition <ref>. The latter provides an asymptotic formula for the number of _q-valued points 𝐚∈^m+1(_q) for which each of the elements L_1(ℱ_𝐚),...,L_n(ℱ_𝐚) generates a prime ideal in R. The formulation of Proposition <ref> should be viewed as an explicit Chebotarev theorem. Our proof makes use of <cit.> and <cit.>.


Factorization types and general counting argument
		Suppose given an _q-rational point a∈^m+1(_q). If R/(L_i(ℱ_𝐚)) is a separable _q-algebra, then because R is a Dedekind domain, the ideal (L_i(ℱ_𝐚))⊂ R admits a prime factorization
		
    (L_i(ℱ_𝐚))
    		 = 𝔣_i1⋯ 𝔣_iℓ_i,
 
such that each residue field κ(𝔣_ij)=R/(𝔣_ij) is a separable extension of _q. The conditions on f_0 and σ_i guarantee that in this case
		
    k_i =_ div(L_i(ℱ_𝐚))|_C\ E = _ div(𝔣_i1)|_C\ E+⋯ +_ div(𝔣_iℓ_i)|_C\ E.

	
Given an _q-rational point a∈^m+1(_q), if R/(L_i(ℱ_𝐚)) is a separable _q-algebra, the factorization type λ_i, 𝐚 of L_i(ℱ_𝐚) is the partition of k_i given in (<ref>).
	
	The n-tuple factorization type counting function of ℒ=(L_1,...,L_n) for a fixed n-tuple λ=(λ_1,...,λ_n), where each λ_i is a partition of k_i= L_i(f_0), is the assignment π_C,ℒ(-;λ) taking the short interval I(f_0,E) to the value
	  
    π_C,ℒ(I(f_0,E);λ)
    	      #{𝐚∈^m+1(_q):R/(L_i(ℱ_𝐚))λ_i,a=λ_i 1≤ i≤ n
    	  }.


Given a positive integer N and a permutation τ∈ S_N, the partition type of τ, denoted λ_τ, is the partition of N determined by the cycle decomposition of τ. Having fixed a subgroup G⊆ S_N, for each partition λ of N we define
	
    P(λ)
    	 _def= #{τ∈ G|λ_τ=λ}/|G|.

	In other words, P(λ) is the probability that a given permutation in G has partition type λ.



	
	Given two positive integers N_1 and N_2, subgroups G_1⊆ S_N_1 and G_2⊆ S_N_2, and partitions λ_1 of N_1 and λ_2 of N_2, we write P(λ_1) and P(λ_2) for the respective probabilities, without explicit reference to the groups G_1 and G_2.


	Define
	
    ℒ(ℱ_𝐀)
    	  
    	L_1(ℱ_𝐀)⋯ L_n(ℱ_𝐀)
    	   ∈   
    	R[𝐀].

Let G=(ℒ(ℱ_𝐀), _q(𝐀)), and fix a partition λ=(λ_1,...,λ_n) of
	
    N
    	  _ div(L_1(f_0)⋯ L_n(f_0))|_C\ E.

Then under the assumptions in <ref>, there exists a constant c=c(B) such that
	
    | π_C,ℒ(ℱ_𝐀)((I(f_0,E);λ)    - P(λ)q^m+1|
    	 ≤ 
    	c q^m+1/2.




	Consider the _q-scheme V(ℒ(ℱ_𝐀)). By <cit.>, for each 1≤ i≤ n there exist an affine open U_i=𝒜_i ⊂𝔸^m+1 and a monic separable polynomial 𝒢_i(t)∈𝒜_i[t] such that 
	
    V(L_i(ℱ_𝐀))_U_i ≅ 𝒜_i[t]/(𝒢_i(t)).

Hence there exists an affine subscheme U=Spec_ 𝒜⊂⋂_i=1^n U_i such that
    V(𝒢(t))_U  
    	V(ℒ(ℱ_𝐀))_U,
    	            𝒢(t)
    	  𝒢_1(t)⋯𝒢_n(t).




		Let U=𝒜 and 𝒢(t)∈𝒜[t] as in Remark <ref>. Then there exists some element a∈_q(𝐀)^× such that a𝒢(t)∈_q[𝐀][t]. Since V(𝒢(t)) V(ℒ(ℱ_𝐀))
		over U, we have that Gal(a𝒢(t),_q(𝐀)) G. Thus by <cit.>, Proposition <ref> holds with a𝒢(t) in place of ℒ(ℱ_𝐀) for some constant c_1(B). Interpreting the closed complement Z𝔸^m+1\ U as an m-cycle in 𝔸^m+1, <cit.> implies that there exists some constant c_2(B) such that #Z(_q)≤ c_2(B)q^m. Finally, 
		
    [ | π_C,ℒ(ℱ_𝐀)(I(f_0,E);λ)-P(λ) q^m+1|                                    ≤            c_1(B) q^m+1/2+c_2(B) q^m;                                                                         ≤                        c(B) q^m+1/2, ]

		where c(B)=c_1(B)+c_2(B).
	




Proof of Theorem <ref>
	
	We obtain Theorem <ref> as a specific case of the following more general Theorem <ref>, which deals with arbitrary factorization types.

In the conditions and notation of Theorem <ref>, let λ be a partition as in Proposition <ref>. Then
	
    π_C,ℒ(ℱ_𝐀)( I(f_0,E);λ) = P(λ_1)⋯ P(λ_n) #I(f_0,E) (1 + O_B(q^-1/2)   ).



	By Proposition <ref>, (ℒ(ℱ_𝐀),_q(𝐀)) S_k_1×⋯× S_k_n. Since P(λ)=P(λ_1)⋯ P(λ_n) and #I(f_0,E)=q^m+1, Proposition <ref> gives 
		
    [                        π_C,ℒ(ℱ_𝐀)( I(f_0,E);λ)                                              =             P(λ_1)⋯ P(λ_n) q^m+1 +O_B(q^m+1/2);                                                                                             = P(λ_1)⋯ P(λ_n) #I(f_0,E) (1 + O_B(q^-1/2)   ), ]

as desired.



	In Theorem <ref>, take each λ_i to be the partition of k_i into a single cell. Then, P(λ_i)=1/k_i=1/(f_0+σ_i) and so
	
    π_C,ℒ(ℱ_𝐀)( I(f_0,E);λ)
    	 = 
    	P(λ_1)⋯ P(λ_n) q^m+1 +O_B(q^m+1/2)

	
    = #I(f_0,E)/∏_i=1^n_ div(f_0+σ_i)|_C\ E (1 + O_B(q^-1/2)   ),

	as desired.



	In the formulation of the Hardy-Littlewood Conjecture <ref>, one can consider polynomial functions more general than the monic linear functions L_i(X)=σ_i+X∈𝒪_K[X]. Some of the resulting variant forms of Conjecture <ref> are well known conjectures or results in their own right. For example, Conjecture <ref> becomes the quantitative Goldbach conjecture if we set n=2 with L_1(X)=X and L_2(X)=σ -X, for σ∈𝒪_K. If one takes L_1(X)= X and L_2(X)=σ_0 + σ_1 X for σ_0,σ_1∈𝒪_K, then Conjecture <ref> provides an asymptotic count of primes in an arithmetic progression.
	
	 Over _q(t), Bary-Soroker and the first author establish a version of Theorem <ref> for non-monic linear functions σ_i1X+σ_i0∈_q[t][X]. We expect that this and other interesting and important variants of Theorem <ref> hold on curves of higher genus over _q. We plan to investigate these questions in future work.










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