The Variance of the Number of Prime Polynomials in Short Intervals and in Residue Classes

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1 International Mathematics Research Notices Advance Access published October 6, 202 J. P. Keating and Z. Rudnick 202) Variance of the Number of Prime Polynomials, International Mathematics Research Notices, rns220, 30 pages. doi:0.093/imrn/rns220 The Variance of the Number of Prime Polynomials in Short Intervals and in Residue Classes Jonathan P. Keating and Zeév Rudnick 2 School of Mathematics, University of Bristol, Bristol BS8 TW, UK and 2 Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel Correspondence to be sent to: rudnick@post.tau.ac.il We resolve a function field version of two conjectures concerning the variance of the number of primes in short intervals Goldston and Montgomery) and in arithmetic progressions Hooley). A crucial ingredient in our work is the recent equidistribution results of N. Katz. Introduction In this note, we study a function field version of two outstanding problems in classical Prime Number Theory, concerning the variance of the number of primes in short intervals and in arithmetic progressions.. Problem : Primes in short intervals The prime number theorem PNT) asserts that the number πx) of primes up to x is asymptotically Lix) = x dt 2. Equivalently, defining the von Mangoldt function as log t Λn) = log p if n= p k is a prime power, and 0 otherwise, then PNT is equivalent to the assertion that ψx) := n x Λn) x as x..) Received April 23, 202; Revised September 9, 202; Accepted September 0, 202 c The Authors) 202. Published by Oxford University Press. All rights reserved. For permissions, please journals.permissions@oup.com.

2 2 J. P. Keating and Z. Rudnick To study the distribution of primes in short intervals, we define, for H x, ψx; H) := Λn)..2) n [x H 2,x+ H 2 ] The Riemann Hypothesis RH) guarantees an asymptotic formula ψx; H) H as long as H > X 2 +o). To understand the behavior in shorter intervals, Goldston and Montgomery [5] studied the variance of ψx; H) and showed conditionally that for X δ < H < X δ, X X 2 ψx; H) H 2 dx Hlog X log H),.3) assuming the RH and the strong ) pair correlation conjecture. Furthermore, they showed that under RH.3) and the strong pair correlation conjecture are in fact equivalent. At this time.3) is still open..2 Problem 2: Primes in arithmetic progressions The PNT for arithmetic progression states that for a modulus Q and A coprime to Q, the number of primes p X with p= A mod Q is asymptotically πx)/φq), where πx) is the number of primes up to X and φq) is the Euler totient function, giving the number of reduced residues modulo Q. Equivalently, if ψx; Q, A) := n X n=a mod Q Λn),.4) then PNT for arithmetic progressions states that for a fixed modulus Q, ψx; Q, A) X φq) as X..5) In most arithmetic applications, it is crucial to allow the modulus to grow with X. Thus, the remainder term in.5) is of the essence. For very large moduli Q > X, there can be at most one prime in the arithmetic progression P = A mod Q so that the interesting range is Q < X. Assuming the Generalized Riemann Hypothesis GRH) gives.5) forq < X /2 o).

3 Variance of the Number of Prime Polynomials 3 The fluctuations of ψx; Q, A) have been studied over several decades, notably allowing also averaging over the modulus Q. Thus, define GX, Q) = A mod Q gcda,q)= ψx; Q, A) X φq) 2.6) and HX, Q) = Q Q GX, Q )..7) The study of the sum HX, Q) has a long history, going under the name of theorems of Barban Davenport Halberstam-type. Among other results is the one due to Montgomery [3] and Hooley [7] asserting that for X/log X) A < Q < X one has for all A> 0, where HX, Q) = QX log Q cqx + O Q 5/4 X 3/4 X 2 ) +,.8) log X) A c = γ + log2π)+ + p log p pp )..9) Hooley [8] showed that assuming GRH,.8) holdsforx /2+ɛ < Q < X with remainder OX 2 /log X) A ). The individual variance GX, Q) is much less understood. Hooley [6] conjectured that under some unspecified) conditions, Friedlander and Goldston [4] show that in the range Q > X, GX, Q) X log Q..0) GX, Q) = X log X X X2 φq) + O X log X) A ) + Olog Q) 3 )..) Note that in this range, there is at most one integer n= A mod Q with n< X. They conjecture that.0) holdsif X /2+ɛ < Q < X.2)

4 4 J. P. Keating and Z. Rudnick and further conjecture that if X /2+ɛ < Q < X ɛ,then GX, Q) = X log Q X γ + log 2π + p Q log p + ox)..3) p They show that both.0) in the range X /2+ɛ < Q < X) and.3) in the range X /2+ɛ < Q < X ɛ ) hold assuming a Hardy Littlewood conjecture with small remainders. For Q < X /2 very little seems to be known. Hooley addresses this in paper V of his series of papers on the subject [9], which he opens by stating An interesting anomaly in the theory of primes is presented by the situation in which known forms of the PNT for arithmetic progressions are only valid for relatively) small values of the common difference k, whereas the theorems of Barban Davenport Halberstam type discussed in I, II, IV are only fully significant for the relatively) larger values of k. The most striking illustration of this contrast is perhaps provided by the conditional theorems at present available on the extended Riemann hypothesis, the ranges of significance of the PNT and of the Barban-Montgomery theorem given in II being then, respectively, k < x /2 ɛ and k > x /2+ɛ.... it is therefore certainly desirable to elicit further forms of the Barban Davenport Halberstam theorem that should be valid for the smaller values of k. In the above Hooley s k corresponds to our Q and x to X, and the roman numerals refer to previous papers in the series written by him. Concerning Conjectures.0) and.3) forgx, Q), Friedlander and Goldston say [4, p. 35] It may well be that these also hold for smaller Q, but below X = Q /2 we are somewhat skeptical. In this paper, we resolve the function field versions of Conjectures.3)and.0), indicating that.0) should hold all the way down to Q > X ɛ. A crucial ingredient in our work are recent equidistribution results of Katz [, 2] described in Sections 4 and 5. 2 Results for Function Fields Let F q be a finite field of q elements and F q [T] the ring of polynomials with coefficients in F q.letp n ={f F q [T]:deg f = n} be the set of polynomials of degree n and M n P n the subset of monic polynomials.

5 Variance of the Number of Prime Polynomials 5 The von Mangoldt function in this case is defined as ΛN) = deg P,ifN = cp k with P an irreducible monic polynomial, and c F q,andλn) = 0 otherwise. The Prime Polynomial Theorem in this context is the identity f M n Λ f) = q n. 2.) 2. Short intervals For A P n of degree n, andh < n, we define short intervals I A; h) :={f : f A q h }=A + P h, 2.2) where the norm of a polynomial 0 f F q [T] is f := q deg f 2.3) and P h ={0} P m 2.4) 0 m h is the space of polynomials of degree at most h including 0). We have #I A; h) = q h+. 2.5) Note: For h < n, if f A q h,thenamonic if and only if f is monic. Hence for A monic, I A; h) consists of only monic polynomials and all monic f s of degree n are contained in one of the intervals I A; h) with A monic. We define, for h < n and A P n, νa; h) = f I A;h) f0) 0 Λ f) 2.6) to be the number of prime powers co-prime to T in the interval I A; h), weighted by the degree of the corresponding prime.

6 6 J. P. Keating and Z. Rudnick We will show in Lemma 4.3 that the mean value of νa; h) when we average over monic A M n is Our goal is to compute the variance ν ; h) := νa; h) = q h+ q n A M n Varν ; h) = νa; h) ν ; h) 2 q n A M n q n ). 2.7) in the limit q. Theorem 2.. Let h < n 3. Then lim q Varν ; h)) = n h ) qh+ We may compare 2.8) with.3), if we make the dictionary X q n, H q h+, log X n, log H h +, 2.9) the conclusion being that Theorem 2. is precisely the analog of the conditional result.3) of Goldston and Montgomery. 2.2 Arithmetic progressions Our second result concerns the analog of the conjectures of Hooley.0) and Friedlander Goldston.3) and allows us to make a definite conjecture in that case. For a polynomial Q F q [T] of positive degree, and A F q [T] coprime to Q and any n> 0, set Ψn; Q, A) = ΛN) 2.0) N M n N=A mod Q the sum over monic polynomials). The Prime Polynomial Theorem in arithmetic progressions states that as n, Ψn; Q, A) qn ΦQ), 2.)

7 Variance of the Number of Prime Polynomials 7 where ΦQ) is the Euler totient function for this context, namely the number of reduced residue classes modulo Q. Now set Gn; Q) = A mod Q gcda,q)= Ψn; Q, A) qn 2 ΦQ). 2.2) We wish to show an analog of Conjecture.0) in the limit of large finite field size, that is, q. Theorem 2.2. and n< deg Q. Then i) Given a finite field F q,letq F q [T] be a polynomial of positive degree, Gn; Q) = nq n where the implied constant is absolute. q2n ΦQ) + On2 q n/2 ) + Odeg Q) 2 ), 2.3) ii) Fix n 2. Given a sequence of finite fields F q and square-free polynomials QT) F q [T] of positive degree with n deg Q, then as q, Gn; Q) q n deg Q ). 2.4) We can compare 2.4) with.0) in the range.2), if we make the dictionary Q Q =q deg Q, log Q deg Q, X q n, log X n. 2.5) The result.) in the range Q > X corresponds to n< deg Q, and the range X /2 < Q < X of.2) corresponds to deg Q < n< 2degQ, so that we recover the function field version of conjecture.0). Note that 2.4) holds for all n, not just that range. Thus, Conjecture.0) may well be valid for all Q > X ɛ. 3 Background on Characters and L-functions We review some standard background concerning Dirichlet L-functions for the rational function field; see, for example, [5, 6].

8 8 J. P. Keating and Z. Rudnick 3. The prime polynomial theorem Let F q be a finite field of q elements and F q [T] the polynomials over F. The zeta function Zu) of F q [T] is Zu) := P u deg P ) 3.) where the product is over all monic irreducible polynomials in F q [T]. The product is absolutely convergent for u < /q. By unique factorization into irreducibles in F q [T], we have for u < /q, Zu) = qu. 3.2) Taking the logarithmic derivative of 3.) and3.2) leads to the Explicit formula Ψn) := N M n ΛN) = q n 3.3) from which we immediately deduce the Prime Polynomial Theorem, for the number πn) of monic irreducible polynomials of degree n: Lemma 3.. πn) = qn n + Oqn/2 ). 3.4) N M n ΛN) 2 = nq n + On 2 q n/2 ), 3.5) where the implied constant is absolute independent of q and n). Proof. We start with the Explicit Formula 3.3) dπd) = q m 3.6) d m and hence mπm) q m. 3.7)

9 Variance of the Number of Prime Polynomials 9 Now and hence q n = d n dπd) = nq n + d n d<n dπd) 3.8) πn) = qn n + Oqn/2 ). 3.9) Likewise with remainder term bounded by d n d<n ΛN) 2 = d 2 πd) = n 2 πn) + d 2 πd) 3.0) N M n d n d n d<n d 2 πd) d 2 πd) d n/2 Inserting 3.9) into3.0) gives the claim. 3.2 Dirichlet characters d n/2 nq d/2 nq qn/2 q < 2nqn/2. 3.) For a polynomial Qx) F q [T] of positive degree, we denote by ΦQ) the order of the group F q [T]/Q)) of invertible residues modulo Q. A Dirichlet character modulo Q is a homomorphism χ : F q [T]/Q)) C, that is, after extending χ to vanish on polynomials which are not coprime to Q, we require χfg) = χf)χg) for all f, g F q [T], χ) = andχf + hq) = χf) for all f, h F q [T]. The number of Dirichlet characters modulo Q is ΦQ). The orthogonality relations for Dirichlet characters are ΦQ) χ mod Q N = A mod Q, χa)χn) = 0 otherwise, 3.2) where the sum is over all Dirichlet characters mod Q and A is coprime to Q, and ΦQ) A mod Q χ = χ 2, χ A) χ 2 A) = 0 otherwise. 3.3)

10 0 J. P. Keating and Z. Rudnick A Dirichlet character χ is even if χcf) = χf ) for 0 c F q. This is in analogy to the number field case, where a Dirichlet character is called even if χ ) =+, and odd if χ ) =. The number Φ ev Q) of even characters modulo Q is Φ ev Q) = ΦQ). 3.4) q We require the following orthogonality relations for even Dirichlet characters. Lemma 3.2. even. Then Proof. Let χ and χ 2 be Dirichlet characters modulo T m, m >. Suppose χ χ 2 is q m B mod T m B0)= We start with the standard orthogonality relation ΦT m ) χ B)χ 2 B) = δ χ,χ ) B mod T m χ B)χ 2 B) = δ χ,χ ) The only nonzero contributions in the sum are those B with B0) 0 equivalently coprime to T m ). We can write each such B uniquely as B = cb,withb 0) =. Since χ χ 2 is even, we have and hence χ χ 2 cb ) = χ χ 2 B ) 3.7) B mod T m χ B)χ 2 B) = q ) B mod T m B0)= χ B)χ 2 B). 3.8) Comparing with 3.6) and using ΦT m ) = q )q m gives the required result. 3.3 Primitive characters A character is primitive if there is no proper divisor Q Q so that χf ) = whenever F is coprime to Q and F = modq. Denoting by Φ prim Q) the number of primitive characters

11 Variance of the Number of Prime Polynomials modulo Q, we clearly have ΦQ) = D Q Φ primd) and hence by Möbius inversion, Φ prim Q) = D Q μd)φ ) Q D 3.9) the sum over all monic polynomials dividing Q. Therefore, Φ prim Q) ΦQ) 2deg Q q. 3.20) Hence as q, almost all characters are primitive in the sense that Φ prim Q) ΦQ) = + O the implied constant depending only on deg Q. ), 3.2) q Likewise, the number Φprim ev Q) of primitive even characters is given by Φprim ev Q) = ) Q μd)φ ev = D q D Q For instance, for QT) = T m, m 2, we find The number Φ prim odd Q) of odd primitive characters is then μd)φ D Q ) Q. 3.22) D Φ ev prim T m ) = q m 2 q ). 3.23) Φprim odd Q) = Φ primq) Φprim ev Q) = ) Φ prim Q) 3.24) q and hence we find that as q with deg Q fixed, almost all characters are primitive and odd: Φprim odd Q) = + O ΦQ) the implied constant depending only on deg Q. ), 3.25) q

12 2 J. P. Keating and Z. Rudnick 3.4 L-functions The L-function Lu,χ)attached to χ is defined as Lu,χ)= P Q χp )u deg P ), 3.26) where the product is over all monic irreducible polynomials in F q [T]. The product is absolutely convergent for u < /q. Ifχ = χ 0 is the trivial character modulo q, then Lu,χ 0 ) = Zu) P Q u deg P ). 3.27) The basic fact about Lu,χ)is that if Q F q [T] is a polynomial of degree deg Q 2, and χ χ 0 a nontrivial character mod Q, then the L-function Lu,χ) is a polynomial in u of degree deg Q. Moreover, if χ is an even character, that is, χcf) = χf ) for 0 c F q,then there is a trivial zero at u= : L,χ)= 0 and hence where P u,χ)is a polynomial of degree deg Q 2. We may factor Lu,χ)in terms of the inverse roots Lu,χ)= u)p u,χ), 3.28) deg Q Lu,χ)= α j χ)u). 3.29) j= The Riemann Hypothesis, proved by Andre Weil in general, is that for each nonzero) inverse root, either α j χ) = or α j χ) =q / ) We define Ψn,χ):= Λ f)χf), 3.3) deg f=n

13 Variance of the Number of Prime Polynomials 3 the sum over monic polynomials of degree n. Taking logarithmic derivative of the L- function gives a formula for Ψn,χ)in terms of the inverse roots α j χ): Ifχ χ 0 is nontrivial, then Weil s theorem 3.30) gives for n> 0 deg Q Ψn,χ)= α j χ) n. 3.32) j= 3.5 The unitarized Frobenius matrix Ψn,χ) deg Q )q n/2, χ χ ) We may state the results in cleaner form if we assume that χ is a primitive character modulo Q. We also define χ even, λ χ := 0 otherwise. 3.34) Then for Q F q [T] a polynomial of degree 2, and χ a primitive Dirichlet character modulo Q, is a polynomial of degree so that L u,χ)= N j= α jχ)u) and L u,χ):= λ χ u) Lu,χ) N = deg Q λ χ 3.35) α j = q j =,...,N. 3.36) For a primitive character modulo Q, we write the inverse roots as α j = q /2 e iθ j and the completed L-function L u,χ)as L u,χ)= deti uq /2 Θ χ ), Θ χ = diage iθ,...,e iθ N ). 3.37) The unitary matrix Θ χ or rather, the conjugacy class of unitary matrices) is called the unitarized Frobenius matrix of χ.

14 4 J. P. Keating and Z. Rudnick Taking the logarithmic derivative of 3.37) we obtain an explicit formula for primitive characters: Ψn,χ)= q n/2 trθχ n λ χ. 3.38) 4 Prime Polynomials in Short Intervals In this section, we prove Theorem 2., the analog of the Goldston Montgomery result.3). 4. An involution For 0 f F q [T], we define f T) := T deg f f ) T or if ft) = f 0 + f T + + f n T n, n= deg f so that f n 0), then f is the reversed polynomial 4.) f T) = f 0 T n + f T n + + f n. 4.2) We also set 0 = 0. Note that f 0) 0and f0) 0 if and only if deg f = deg f. Moreover restricted to polynomials which do not vanish at 0, equivalently are co-prime to T, then is an involution: We also have multiplicativity: f = f, f0) ) fg) = f g. 4.4) Lemma 4.. For f P n with f0) 0, we have Λ f ) = Λ f). Proof. For polynomials which do not vanish at 0, that is, are co-prime to T, P is irreducible if and only if P is irreducible. This is because if P = AB with A, B of positive degree, then P = AB) = A B and if P 0) 0, then the same holds for A, B and then deg A = deg A> 0, deg B = deg B > 0soP is reducible; applying again and using that it is an involution since P 0) 0) gives the reverse implication.

15 Variance of the Number of Prime Polynomials A fundamental relation We can now express the number of primes in our short intervals in terms of the number of primes in a suitable arithmetic progression. Define Ψn; Q, A) = f Pn f=a mod Q Λ f), 4.5) the sum over all polynomials of degree n, not necessarily monic. Lemma 4.2. For B P n h, νt h+ B; h) = Ψn; T n h, B ). 4.6) Proof. Let B P n h.wehave f = T h+ B + g I T h+ B; h), g P h if and only if f = B + T n h g, and thus we find f I T h+ B; h) f B mod T n h. 4.7) As f runs over I T h+ B; h) with the proviso that f0) 0, f runs over all polynomials of degree exactly n satisfying f B mod T n h, and for these Λ f) = Λ f ). 4.3 Averaging We want to compute the mean value and variance of νa, h). To perform the average over A, note that every monic polynomial f M n can be written uniquely as f = T h+ B + g, B M n h+), g P h. 4.8) We therefore can decompose M n as the disjoint union of intervals I T h+ B; h) parameterized by B M n h+) : M n = I T h+ B; h). 4.9) B M n h+) To compute averages ν on short intervals, it suffices, by the foregoing, to take A= T h+ B and to average over all B M n h+).

16 6 J. P. Keating and Z. Rudnick The map gives a bijection : M n h+) {B P n h ) : B 0) = }, 4.0) B B with polynomials of degree n h + ) with constant term. Thus, as B ranges over M n h+), B ranges over F q [T]/T n h )), all invertible residue class mod T n h so that B 0) =. Thus, the mean value is and the variance is 4.4 The mean value ν ; h) = Varν ; h)) = #M n h = q n h B M n h νt h+ B;, h) B mod T n h B 0)= #M n h = q n h Ψn; T n h, B ) 4.) B M n h νt h+ B;, h) ν 2 B mod T n h B 0)= Ψn; T n h, B ) ν ) The computation of the mean value ν ; h) = q n A M n νa; h) is a simple consequence of the Prime Polynomial Theorem. The result is the following. Lemma 4.3. Let 0 < h < n. The mean value of νa, ; h) is ν ; h) =q h+ q n ). 4.3) Proof. We do the computation in two different ways as a check of the all-important relation 4.6). By using the definition of ν, weobtain ν ; h) = #M n h B M n h f I T h+ B;h) f0) 0 Λ f)

17 = #M n h Variance of the Number of Prime Polynomials 7 Λ f) ΛT n ). 4.4) f M n Note that #M n h = q n h = ΦT n h ) q. 4.5) Using 4.6), the mean value of ν ; h) is Hence ν ; h) = = = ΦT n h ) ΦT n h ) ΦT n h ) B mod T n h B 0)= deg f =n f 0)= Λ f ) Ψn; T n h, B ) deg f =nλ f ) c F q ΛcT n ) = Λ f ) ΛT n ). 4.6) q n h f M n ν ; h) = q n h qn ) = q h+ q ) n on using the Prime Polynomial Theorem in the form 3.3). 4.5 An alternate expression for νa; h) 4.7) Using the standard orthogonality relation 3.6) for Dirichlet characters modulo T n h gives an alternate expression for Ψn; T n h, B ) and hence for νt h+ B; h): Ψn; T n h, B ) = ΦT n h ) χ mod T n h χb ) deg f =n Λ f )χ f ). 4.8) Only even characters give a nonzero term, because Λcf) = Λ f) for c F q,and each even character contributes a term χb q ) ΦT n h ) deg f=n monic Λ f)χ f) = χb ) Ψn,χ), 4.9) qn h

18 8 J. P. Keating and Z. Rudnick where Ψn,χ)= deg f=n monic Λ f)χf). 4.20) Note that the number of even characters mod T n h is exactly q ΦT n h ) = q n h. The trivial character χ 0 contributes the term q )q n ) ΦT n h ) = q h+ q n ) = ν. 4.2) Thus, we find that the difference between νt h+ B; h) and its mean ν is 4.6 The variance νt h+ B; h) ν = q n h Our result here is the following. Theorem 4.4. χ χ 0 mod T n h even χb )Ψ n,χ). 4.22) Fix n> 0andlet0< h < n. Asq, the variance of ν is given by Varν) = q h+ q n h χ ) ) n h trθχ n 2 + O q + n2, 4.23) n/2 q where the sum is over primitive even characters modulo T n h, the implied constant depending only on n. Proof. By 4.22), we have Varν) = q n h B mod T n h B 0)= q 2n h ) 2 χb )Ψ n,χ). 4.24) χ χ 0 even Expanding the sum over characters, and interchanging the order of summation to use the orthogonality relation of Lemma 3.2 gives Varν) = q 2n h ) χ χ 0 even Ψn,χ) )

19 Variance of the Number of Prime Polynomials 9 There are altogether ϕt n h )/q ) = q n h even characters modulo T n h,of which Oq n h 2 ) are nonprimitive. We bound the contribution of the nontrivial nonprimitive characters by Ψn,χ)= Onq n/2 ) via the RH. Thus, the nonprimitive characters contribute a total of On 2 q h ) to Varν). Using the explicit formula 3.38) for primitive even characters and the RH gives Ψn,χ) 2 = q n trθ n χ 2 + On h)q n/2 ). 4.26) Therefore, Varν) = q h+ q n h χ ) ) n h trθχ n 2 + O q + n2, 4.27) n/2 q where the sum is over primitive even characters modulo T n h, whose number is q n h q ). 4.7 Proof of Theorem 2. Thus, we found that for h < n 3, the variance of ν is given by Varν) = ) ) n h trθ n qh+ χ q 2 +O q + n2 n/2 q 4.28) with trθ n χ 2 being the mean value of trθ n χ 2 over the set of all primitive even Dirichlet characters modulo T n h. Thus, as q,varν)/q h+ is asymptotically equal to the form factor trθ n χ 2. Theorem 4.5. To proceed further, we need to invoke a recent result of Katz [2]: [2, Theorem.2] Fix m 3. The unitarized Frobenii Θ χ for the family of even primitive characters mod T m+ become equidistributed in the projective unitary group PUm ) of size m, as q. Applying Theorem 4.5 gives lim q q n h q ) χ trθχ n 2 = tru n 2 du. 4.29) PUn h 2)

20 20 J. P. Keating and Z. Rudnick We may pass from the projective unitary group PUn h 2) to the unitary group because the function tru n 2 being averaged is invariant under scalar multiplication. As is well known see, e.g., [3]), for n> 0, tru n 2 du = minn, N). 4.30) UN) Therefore, we find This concludes the proof of Theorem Prime Polynomials in Arithmetic Progressions Varν) q h+ n h 2), q. 4.3) In this section, we prove Theorem 2.2, giving the function field analog of the conjectures of Hooley.0) and Friedlander Goldston.3). 5. The range n< deg Q We prove the result in the range n< deg Q by elementary arguments: Proposition 5.. For 0 < n< deg Q, we have Gn; Q) = nq n where the implied constant is independent of q, n, and Q. q2n ΦQ) + On2 q n/2 ) + Odeg Q) 2 ), 5.) Proof. Assume as we may that deg A< deg Q. Ifn< deg Q then the only solution to the congruence N = A mod Q, withdegn = n< deg Q is A if deg A= n) or else there is no solution. Therefore, if n< deg Q, then ΛA) A is monic and deg A= n, Ψn; Q, A) = 0 otherwise. 5.2)

21 Variance of the Number of Prime Polynomials 2 Thus, Gn; Q) = gcda,q)= = deg A=n A monic gcda,q)= q n ΦQ) ΛA) A is monic and deg A= n 0 otherwise qn ΛA) 2 2 ΦQ) deg A=n A monic gcda,q)= ΛA) + q2n ΦQ). 2 By the Prime Polynomial Theorem 3.3), deg A=n A monic gcda,q)= According to Lemma 3., and so we find deg A=n A monic gcda,q)= Gn; Q) = nq n Since for n< deg Q, q n ΦQ) q P Q prime ΛA) = q n ΛA) 2 = P Q prime deg P n deg P = q n + Odeg Q). 5.3) deg A=nΛA) 2 P Q deg P n deg P ) 2 = nq n + On 2 q n/2 ) + Odeg Q) 2 ) 5.4) q2n q ΦQ) + On2 q n/2 ) + Odeg Q) 2 n ) ) + O ΦQ) deg Q. 5.5) ) P q deg P deg Q prime ) deg Q, 5.6) P q we find as claimed. Gn; Q) = nq n q2n ΦQ) + On2 q n/2 ) + Odeg Q) 2 ) 5.7)

22 22 J. P. Keating and Z. Rudnick 5.2 The range n deg Q To deal with the range n deg Q we relate the problem to an equidistribution statement for the unitarized Frobenii of primitive odd characters. It transpires that Gn; Q) is related to the mean value of the modulus squared of the trace of the Frobenius matrices associated with the family of Dirichlet L-functions for characters modulo Q: Theorem 5.2. Fix n and let Q F q [T] have degree deg Q 2. Then Gn; Q) = trθ n q n χ 2 + ) + O q deg Q) 2 where denotes the average over all odd primitive characters modulo Q. Proof. The orthogonality relation 3.2) gives Ψn; Q, A) = ΦQ) = ΦQ) χ mod Q χ mod Q The trivial character χ 0 gives a contribution of Hence, ΦQ) Ψn; Q, A) deg N=n gcdn,q)= ΛN) = qn ΦQ) = ΦQ) χa) deg N=n q χn)λn) ), 5.8) χa)ψ n,χ). 5.9) qn ΦQ) ΦQ) P Q deg P n P Q deg P n deg P + ΦQ) deg P. 5.0) χ χ 0 χa)ψ n,χ). 5.) We square out and average over all A mod Q coprime with Q. Using the orthogonality relation 3.3) gives Gn; Q) = ΦQ) Ψn,χ) 2 + ΦQ) χ χ 0 P Q deg P n 2 deg P. 5.2)

23 Variance of the Number of Prime Polynomials 23 For nontrivial characters which are either even or imprimitive, we use the RH 3.33) to bound Ψn,χ) 2 q n deg Q ) 2. Therefore, we find Gn; Q) = ΦQ) χ primitive, odd + O q n deg Q) Ψn,χ) 2 2 #{χ either even or imprimitive} ΦQ) ). 5.3) The number of even characters is ΦQ)/q ), and the number of imprimitive characters is Oq n deg Q) 2 ). and therefore OΦQ)/q). Hence the remainder term above is bounded by For each primitive odd character, the explicit formula 3.38) says Gn; Q) = q n ΦQ) Ψn,χ)= q n/2 trθ n χ 5.4) χ odd primitive trθ n χ 2 + Oq n deg Q) 2 ). 5.5) Replacing ΦQ) by the number of odd primitive characters times + O ) gives 5.8). q We now use another recent equidistribution result of Katz []: Theorem 5.3 Katz []). F q Fix m 2. Suppose that we are given a sequence of finite fields and square-free polynomials QT) F q [T] of degree m. As q, the conjugacy classes Θ χ with χ running over all primitive odd characters modulo Q, are uniformly distributed in the unitary group Um ). Note that Theorem 5.3 gives a nonstandard form of equidistribution, in that it deals with a family of L-functions which are not parameterized by an algebraic variety. Its proof in [] relies on the recent book [0] which studies such cases. Using Theorem 5.3 we obtain for n> 0, lim q trθn χ 2 = tru n 2 du, 5.6) Udeg Q )

24 24 J. P. Keating and Z. Rudnick where du is the Haar probability measure on the unitary group UN). Since[3] tru n 2 du = minn, N), 5.7) UN) we find Gn; Q) lim = minn, deg Q ), 5.8) q q n which is the statement of Theorem 2.2. Acknowledgements We thank Nick Katz for several discussions, and Julio Andrade and the referees for their comments. Funding J.P.K was supported by a grant from the Leverhulme Trust and by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant number FA The U.S. Government is authorized to reproduce and distribute reprints for Governmental purpose notwithstanding any copyright notation thereon. Z.R. was supported by the Israel Science Foundation grant No. 083/0). Appendix. A Calculation Based on a Hardy Littlewood-Type Conjecture In the number-field setting, the problems we have considered here have previously been explored using the Hardy Littlewood conjecture relating to the density of generalized twin primes [4, 4]. In this appendix, we sketch a heuristic calculation showing how the corresponding conjecture in the function field setting may be used in the same way. As an example, we focus on estimating Gn, Q). The twin prime conjecture of Hardy and Littlewood for the rational function field F q [T] states that, given a polynomial 0 K F q [T], and n> deg K, deg f=n Λ f)λ f + K) SK)q n A.)

25 as q n, where the singular series SK) is given by Variance of the Number of Prime Polynomials 25 SK) = P ) 2 ν ) KP ), A.2) P P with the product involving all monic irreducible P and, P K, ν K P ) = #{A mod P : AA + K) = 0modP }= A.3) 2, P K. While for fixed q and n the problem is currently completely open, for fixed n and q,a.) is known to hold [, 2] forq odd, in the form Λ f)λ f + K) = q n + O n q n 2 ). A.4) deg f=n Note that SK) = + O n q ). We want to use A.) to compute Gn; Q) and to show that the result is consistent with Gn; Q) q n deg Q ), n> deg Q. A.5) It turns out that this can be done if we ignore the contribution from the remainder implicit in A.). The remainder term in A.4) is insufficient for our purposes. Starting with Gn; Q) = gcda,q)= Ψn; Q, a) qn 2 ΦQ), A.6) we have Gn; Q) = gcda,q)= qn Ψn; Q, A) 2 2 ΦQ) gcda,q)= Ψn; Q, A) + q2n ΦQ). A.7)

26 26 J. P. Keating and Z. Rudnick The first moment of Ψn; Q, A) is gcda,q)= Ψn; Q, A) = By Lemma 3., we may safely replace gcda,q)= deg f=n gcd f,q)= = deg f=n Λ f) Λ f) = q n For the second moment of Ψn; Q, A) we have Now gcda,q)= Ψn; Q, A) 2 = deg f=n gcd f,q)= deg P n P Q prime deg f=n deg gcd f,q)>0 deg P. Ψn; Q, A) = q n + negligible. deg f=deg g=n f g mod Q gcd f,q)= = deg f=n gcd f,q)= Λ f)λg) Λ f) 2 + deg f=deg g=n f g mod Q f g gcd f,q)= Λ f) 2 = nq n + On 2 q n/2 ) P Q deg P n Λ f) deg P ) 2. A.8) A.9) Λ f)λg). A.0) A.) For the sum over f g, we write the condition f = g mod Q as g = f + JQ, J 0, deg J < n deg Q the number of such J of degree j is q )q j )andthen deg f=deg g=n f g mod Q f g gcd f,q)= Λ f)λg) = deg J<n deg Q J 0 ψ 2 n; JQ), A.2)

27 Variance of the Number of Prime Polynomials 27 where for K 0, deg K < n, ψ 2 n; K) := deg f=n f monic Λ f)λ f + K). A.3) Clearly, we can split the right-hand side of A.2 as follows: deg f=deg g=n f g mod Q f g gcd f,q)= n deg Q Λ f)λg) = j=0 deg J= j J 0 ψ 2 n; JQ). A.4) The J-sum here is not restricted to monic polynomials. We can restrict it to monics, multiplying by q. Then inserting A.), we have as q n. deg f=deg g=n f g mod Q f g gcd f,q)= n deg Q Λ f)λg) q n q ) j=0 In order to estimate the J-sum in A.5), consider J monic SJQ) J s = α J monic J s where the equality follows from inserting A.2) and Hence, J monic α = P SJQ) = α P J s P 2 P Q P JQ deg J= j J 0 J monic SJQ) A.5) P P 2, A.6) ). A.7) P ) 2 J monic J s P J P Q P P 2. A.8) Since the summand on the right-hand side is multiplicative, we may write this as J monic SJQ) = α P J s P 2 P Q P Q + ) P P s P 2 P Q ). A.9) P s

28 28 J. P. Keating and Z. Rudnick Therefore, J monic SJQ) = αζ J s A s) P P 2 P Q P Q ) + P s P 2) A.20) with ζ A s) = P ). A.2) P s Hence, J monic Furthermore, SJQ) = αζ J s A s) P + P 2 P s P 2) P Q J monic ) P ) +. A.22) P s P 2) SJQ) = αζ J s A s)ζ A s + ) ) P + P 2 P s P 2) P Q P 2 + P s+ P 2) P P 2 P 2s+2 ). A.23) It is convenient to re-express these formulae in terms of the variable u= /q s. Thus, J =u degj, P =u degp,and with J monic SJQ)u degj = αzu)zu/q) P Q P Zu) = P + P udeg P ) P 2 P 2) + 2udeg P P P 2) u 2degP ) P P 2) A.24) u deg P ) = qu. A.25) We can now estimate the J-sum in A.5) by denoting F u) = SJQ)u deg J J monic A.26)

29 Variance of the Number of Prime Polynomials 29 and using deg J= j J 0 J monic SJQ) = F u) du, 2πi u A.27) j+ where the contour is a small circle enclosing the origin but no other singularities of the integrand. Expanding the contour beyond the poles of F u) at u= /q and u= coming from the factors of Zu) and Zu/q) in A.24)), we find that as q where we have used deg J= j J 0 J monic SJQ) q j Q ΦQ) q, P Q P P = Q ΦQ). A.28) A.29) Note that the first term in A.28) coincides after the usual translation with that in the corresponding expression in the number field calculation [4], but that interestingly the second term has a different form. Finally, substituting A.28) intoa.5) and incorporating the estimates for the other terms in A.7), we find that We now observe that as q Gn; Q) q n deg Q Q ). A.30) ΦQ) Q ΦQ) A.3) and so in this limit, when n is fixed with deg Q n+, this calculation matches Theorem 2.2. Furthermore, when deg Q with q fixed we have that Gn; Q) q n deg Q, A.32) which is consistent with the Hooley s conjecture.0) in the number field case.

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