The fourth power mean of Dirichlet L-functions in Fq[T]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {F}}_q [T]$$\end{document}

We prove results on moments of L-functions in the function field setting, where the moment averages are taken over primitive characters of modulus R, where R is a polynomial in Fq[T]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {F}}_{q}[T]$$\end{document}. We consider the behaviour as degR→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{deg}\,}}R \rightarrow \infty $$\end{document} and the cardinality of the finite field is fixed. Specifically, we obtain an exact formula for the second moment provided that R is square-full, an asymptotic formula for the second moment for any R, and an asymptotic formula for the fourth moment for any R. The fourth moment result is a function field analogue of Soundararajan’s result in the number field setting that improved upon a previous result by Heath-Brown. Both the second and fourth moment results extend work done by Tamam in the function field setting who focused on the case where R is prime. As a prerequisite for the fourth moment result, we obtain, for the special case of the divisor function, the function field analogue of Shiu’s generalised Brun–Titchmarsh theorem.


Introduction
The study of moments of families L-functions is a central theme in analytic number theory. These moments are connected to the famous Lindelöf hypothesis for such L-functions and have many applications in analytic number theory. It is a very challenging problem to establish asymptotic formulas for higher moments of families of L-functions and until now we only have asymptotic formulas for the first few moments of any given family of L-functions. However, we do have precise conjectures for higher moments of families of L-functions due to the work of many mathematicians (see for example [2] and [3]). In this paper the focus is on the moments of Dirichlet L-functions associated to primitive Dirichlet characters.
In 1981, Heath-Brown [4] proved that * where for all positive integers q, * χ mod q represents a summation over all primitive Dirichlet characters of modulus q, φ * (q) is the number of primitive characters of modulus q, and ω(q) is the number of distinct prime divisors of q and L(s, χ) is the associated Dirichlet L-function.
In the equation above (1), in order to ensure that the error term is of lower order than the main term, we must restrict q to ω(q) ≤ log log q − 7 log log log q log 2 .
Here, the error terms are of lower order than the main term without the need to have any restriction on q.
In a breakthrough paper, Young [11] obtained explicit lower order terms for the case where q is an odd prime and was able to establish the full polynomial expansion for the fourth moment of the associated Dirichlet L-functions. In other words, he proved that where the constants c i are computable. The error term was subsequently improved by Blomer et al. [1] who proved that In the function field setting Tamam [9] established that Here, Q is an irreducible, monic polynomial in F q [T ] with F q a finite field with q elements; χ 0 is the trivial character (in this case, of modulus Q); and, for non-trivial characters of modulus Q, where M is the set of monic polynomials F q [T ].
In this paper we prove the function field analogue of Soundararajan's fourth moment result, which is also an extension of Tamam's fourth moment result. In order to accomplish this we prove, along the way, a function field analogue of a special case of Shiu's Brun-Titchmarsh theorem for multiplicative functions [7]. We also obtain an asymptotic main term for the second moment. This generalises Tamam's result in that her result is for all primitive characters of prime modulus, whereas our result is for primitive characters of any modulus. Note, however, that Tamam's result is exact. By considering only square-full moduli, we also obtain an exact formula.

Notation and statement of results
Let q ∈ N be a prime-power, not equal to 2. We denote the finite field of order q by F q . We denote the ring of polynomials over the finite field F q by A := F q [T ]. Unless otherwise stated, for a subset S ⊂ A we define S n := {A ∈ S : deg A = n}. We identify A 0 with F q . Also, if we have some non-negative real number x, then range deg A ≤ x is not taken to include the polynomial A = 0.
The norm of A ∈ A\{0} is defined by |A| := q deg A , and for the zero polynomial we define |0| := 0.
We denote the set of monic polynomials in A by M. For a ∈ F * q we denote the set of polynomials, with leading coefficient equal to a, by aM. Because A is an integral domain, an element is prime if and only if it is irreducible. We denote the set of prime monic polynomials in A by P, and all references to primes (or irreducibles) in the function field setting are taken as being monic primes. Also, when indexing, the upper-case letter P always refers to a monic prime. Furthermore, if we range over polynomials E that divide some polynomial F, then these E are taken to be the monic divisors only.
Suppose f , g : D −→ C are functions from the domain D to the complex numbers, where either D ⊆ A or D ⊆ C, and f and/or g may be dependent on q. We take f (x) = O g(x) to mean: There exists a positive constant c such that for all q and all x ∈ D we have | f (x)| ≤ c|g(x)|. Now suppose that we have some variable (not equal to the variable q) taking values in a set E, which f and/or g may depend on. Then, we take f (x) = O g(x) to mean: For each ∈ E, there exists a positive constant c such that for all q and all x ∈ D we have | f (x)| ≤ c |g(x)|. We take f (x) g(x) and g (x) f Due to point 2, we can view a character χ of modulus R as a function on A\RA. This makes expressions such as χ(A −1 ) well-defined for A ∈ A\RA * .
We can deduce that χ(1) = 1 and |χ(A)| = 1 when (A, R) = 1. We say that χ is the trivial character of modulus R if χ(A) = 1 when (A, R) = 1, and this is denoted by χ 0 . Otherwise, we say that χ is non-trivial. Also, there is only one character of modulus 1 and it simply maps all A ∈ A to 1.
It can easily be seen that the set of characters of a fixed modulus R forms an abelian group under multiplication. The identity element is χ 0 . The inverse of χ is χ, which is defined by χ(A) = χ(A) for all A ∈ A. It can be shown that the number of characters of modulus R is φ(R).
A character χ is said to be even if χ(a) = 1 for all a ∈ F * q . Otherwise, we say that it is odd. The set of even characters of modulus R is a subgroup of the set of all characters of modulus R. It can be shown that there are 1 q−1 φ(R) elements in this group.

Definition 2.2 (Primitive Character)
Let R ∈ M, S | R and χ be a character of modulus R. We say that S is an induced modulus of χ if there exists a character χ 1 of modulus S such that χ is said to be primitive if there is no induced modulus of strictly smaller norm than R. Otherwise, χ is said to be non-primitive. φ * (R) denotes the number of primitive characters of modulus R.
We note that all trivial characters of some modulus R = 1 are non-primitive as they are induced by the character of modulus 1. We also note that if R is prime, then the only non-primitive character of modulus R is the trivial character of modulus R. We denote a sum over primitive characters of modulus R by the standard notation * χ mod R .

Definition 2.3 (Dirichlet L-functions)
Let χ be a Dirichlet character. The associated L-function, L(s, χ), is defined for Re(s) > 1 by This has an analytic continuation to either C or C\{1}, depending on the character.
In this paper, we will prove the following three main results.
Furthermore, in order to prove Theorem 2.6 we are required to prove a specific case of the function field analogue of Shiu's generalised Brun-Titchmarsh theorem. This allows us to estimate sums of the form given certain conditions on X , A, G ∈ M and y ≥ 0.

Function field background
We provide some definitions and results relating to function fields that are needed in this paper. Many of these results are well known and so we do not provide a proof. Some proofs can be found in Rosen's book [6], particularly chapter 4.

Definition 3.2 (ω Function)
For all R ∈ A\{0} we define ω(R) to be the number of distinct prime factors of R.

Definition 3.3 (Ω Function)
For all R ∈ A\{0} we define Ω(R) to be the total number of prime factors of R (i.e. counting multiplicity).
It is not hard to show that Similarly, we define p + (R) to be the smallest positive integer such that if P | R then deg P ≤ p + (R).
Recall the Möbius inversion formula tells us that if g, f are functions on M satisfying for all R ∈ M. By applying this to (2) and making use of Lemma 3.6 we obtain the first result. The second result follows similarly to the first.

Corollary 3.8 For all R ∈ M we have that
Proof This follows easily from Lemma 3.7 when we take A, B = 1.
For a character χ we will, on occasion, write the associated L-function as where we define for all non-negative integers n and all characters χ .
Suppose χ is the character of modulus 1 and Re(s) > 1. Then, L(s, χ) is simply the zeta-function for the ring A. That is, We note further that The far-RHS provides a meromorphic extension for ζ A to C with a simple pole at 1. The following Euler product formula will also be useful for Re(s) > 1. Now suppose that χ 0 is the trivial character of some modulus R and Re(s) > 1. It can be shown that So, again, the far-RHS provides a meromorphic extension for L(s, χ 0 ) to C with a simple pole at 1.
Finally, suppose that χ is a non-trivial character of modulus R and Re(s) > 1. It can be shown that This is just a finite polynomial in q −s , and so it provides a holomorphic extension for L(s, χ) to C. Theorem 3.9 (Functional Equation for L-functions of Primitive Characters) Let χ be a primitive character of some modulus R = 1. If χ is even, then L(s, χ) satisfies the function equation and if χ is odd, then L(s, χ) satisfies the function equation where |W (χ )| = 1.
A generalisation of the theorem above appears in Rosen's book [6, Theorem 9.24 A].
That is, and Taking the squared modulus of both sides of (3) and of (4), we see that and By the linear independence of powers of q −s we can see that |L(s, χ)| 2 is equal to the sum of the terms n = 0, 1, . . . , deg R − 1 on the RHS of (5) and the terms n = 0, 1, . . . , deg R − 2 on the RHS of (6). That is, Hence, Proof The functional equation for even primitive characters gives us that For any primitive character χ 1 of modulus R = 1, we define L −1 (χ 1 ) := 0 and recall that L deg R (χ 1 ) = 0. If we define for i = 0, 1, . . . , deg R, then (7) gives us that and Similarly as in the proof of Lemma 3.10, we take the squared modulus of both sides of (8) and (9), and use the linear independence of powers of q −s , to obtain We now take s = 1 2 and simplify to obtain Hence, It is convenient to define

Multiplicative functions on F q [T]
In this section we state and prove some results for the functions μ, φ and ω that are required for the proofs of the main theorems. We will need the following well-known theorem.

Theorem 4.1 (Prime Polynomial Theorem) We have that
where the implied constant is independent of q. We reserve the symbol c for the implied constant.
We will also need the following two definitions.

Definition 4.2 (Radical of a Polynomial, Square-free, and Square-full) For all R ∈ A
we define the radical of R to be the product of all distinct monic prime factors that divide R. It is denoted by rad(R). If R = rad(R), then we say that R is square-free.
If for all P | R we have that P 2 | R, then we say that R is square-full.

Definition 4.3 (Primorial Polynomials)
Let (S i ) i∈Z >0 be a fixed ordering of all the monic irreducibles in A such that deg S i ≤ deg S i+1 for all i ≥ 1 (the order of the irreducibles of a given degree is not of importance in this paper). For all positive integers n we define We will refer to R n as the n-th primorial. For each positive integer n we have unique non-negative integers m n and r n such that where the Q i are distinct monic irreducibles of degree m n + 1. This definition of primorial is not standard. Now, before proceeding to prove results on the growth of the ω and φ functions, we note that for all R ∈ A\{0} . The first equation holds for all s ∈ C. The second holds for all s ∈ C\{0} and is obtained by differentiating the first with respect to s. Also, for all square-full R ∈ A\{0} we have that The first equation holds for all s ∈ C. The second holds for all s ∈ C\{1} and is obtained by differentiating the first with respect to s.

Lemma 4.4 For all positive integers n we have that
Proof By (11) and the prime polynomial theorem, we see that By taking logarithms of both equations above, we deduce that log q log q |R n | = m n + O(1).
Proof It suffices to prove the claim for the primorials. Indeed, if this is true, then taking To prove the middle relation above, we first recall that the prime polynomial theorem m . From this, we can deduce that there is a constant c ∈ (0, 1), which is independent of q, such that #P ≤m ≥ cq m 2 for all positive integers m. In particular, if we take m = 2 log q log n c , then #P ≤m ≥ n . So, where the second relation follows from the prime polynomial theorem again.
The following four results well-known (at least, their analogues in the number field setting are), and their proofs follow the same method as Lemma 4.5 above: Prove the claim for the primorials by using the prime polynomial theorem and perhaps Lemma 4.4, and then generalise to all R ∈ M. Lemma 4. 6 We have that and for infinitely many R ∈ A we have that where a and b are positive constants which are independent of q and R.
and for infinitely many R ∈ A we have that where c and d are positive constants which are independent of q and R.

Note, the fourth result follows easily from the third
We end this section with three more lemmas.

Lemma 4.10
We have that

Lemma 4.11
We have that While it is not a result on multiplicative functions, the proof of the following lemma uses several results from this section.
Proof For all positive integers x we have that By (12), (13), and Lemma 4.5, we see that .
The proof follows.

The second moment
We now proceed to prove Theorems 2.4 and 2.5.
Proof of Theorem 2.4 By using the functional equation for Dirichlet L-functions, we have that For the first term on the RHS, by Lemma 3.7 and Corollary 3.8, we have By Lemma 4.12, we have that For the off-diagonal terms, let us consider the case where deg Hence, Hence, we have that Finally, By similar methods as previously in the proof, we can see that the above is O(1). The result follows.
Proof of Theorem 2. 5 We have that * The second equality follows from Lemma 3.7. For the last equality we note that if R is square-full, E F = R, and μ(E) = 0, then F and R have the same prime factors. Therefore, if we also have that (A, R) = 1 and B ≡ A(mod F), then (B, R) The last equality follows from the fact that F and R have the same prime factors, and so, if Hence, By applying this to (18), and using (12) to (15), we see that *

The Brun-Titchmarsh theorem for the divisor function in F q [T]
In this section we prove a specific case of the function field analogue of the generalised Brun-Titchmarsh theorem. The generalised Brun-Titchmarsh theorem in the number field setting was proved by Shiu [7]. It gives upper bounds for sums over short intervals and arithmetic progressions of certain multiplicative functions. We will look at the case where the multiplicative function is the divisor function in the function field setting.
The main results in this section are the following two theorems.
Theorem 6.1 Suppose α, β are fixed and satisfy 0 < α < 1 2 and 0 < β < 1 2 . Let X ∈ M and y be a positive integer satisfying β deg X < y ≤ deg X. Also, let A ∈ A and G ∈ M satisfy (A, G) = 1 and deg G < (1 − α)y. Then, we have that Intuitively, this seems to be a good upper bound. Indeed, all N in the sum are of degree equal to deg X , and so this suggests that the average value that the divisor function will take is deg X . Also, there are q y 1 |G| ≈ q y 1 φ(G) possible values for N in the sum. Theorem 6.2 Suppose α, β are fixed and satisfy 0 < α < 1 2 and 0 < β < 1 2 . Let X ∈ M and y be a positive integer satisfying β deg X < y ≤ deg X. Also, let A ∈ A and G ∈ M satisfy (A, G) = 1 and deg G < (1 − α)y. Finally, let a ∈ F * q . Then, we have that Our proofs of these two theorems are based on Shiu's proof of the more general theorem in the number field setting [7]. We begin by proving preliminary results that are needed for the main part of the proofs.
The Selberg sieve gives us the following result. A proof is given in [10]. |D| #S D + r (D) and 0 < ω(D) < |D|. Also, define ψ multiplicatively by ψ(P) = |P| ω(P) − 1 and ψ(P e ) = 0 for e ≥ 2. We then have that Proof Let us define Then, we have that which is what we want to bound.
This follows from the fact that K and D are coprime and that deg K where |c D | ≤ 1. Therefore, we have ω(D) = 1 and |r (D)| ≤ 1 for all D | Q z . We also have that ψ(D) = φ(D) for square-free D.
We can now see that and we have that To this we apply Lemma 4.11 and the fact that Also, we have that The result now follows by applying Theorem 6.3.
The proof of the following corollary is almost identical to the proof above. Corollary 6.5 Let X ∈ M and y be a positive integer satisfying y ≤ deg X. Also, let K ∈ M and A ∈ A satisfy (A, K ) = 1. Finally, let z be a positive integer such that deg K + z ≤ y, and let a ∈ F * q . Then, Lemma 6. 6 We have that deg P≤w In particular, we can find an absolute constant d such that deg P≤w Proof By using the prime polynomial theorem, we have that deg P≤w The proof follows by noting that w n=1 q n n 2 = Lemma 6.7 Let 0 < α, β < 1 2 , let z > q be an integer, and let w(z) := log q z. Then, as z → ∞, where d is as in Lemma 6.6. In particular, this implies that (under the condition that z > q).
Proof Let δ > 0. We will optimise on the value of δ later. We have that where the last two relations follow from the Taylor series for the exponential function. Continuing, where the last inequality follows from Lemma 6.6. By using the definition of w(z), we have that and if we take Lemma 6.8 Let z and r be a positive integers satisfying r log q r ≤ z. Then, Proof Let 3 4 ≤ δ < 1. We will optimise on the value of δ later. We have that where the last relation uses the Taylor series for the exponential function.
Note that where the last relation uses the fact that δ ≥ 3 4 . Also, we can write 1 |P| δ = 1 |P| + and that deg P≤ z where the second-to-last relation follows from a similar calculation as (21).

We substitute (20), (21), and (22) into (19) to obtain
We can now take δ = 1 − r log q r 4z (by the conditions on r given in theorem, we have that 3 4 ≤ δ < 1, as required). Then, Proof of Theorem 6. 1 We will need to break the sum into four parts. First, we define z := α 10 y. Now, for any N in the summation range, we can write where deg P 1 ≤ deg P 2 ≤ · · · ≤ deg P n and j ≥ 0 is chosen such that deg P 1 e 1 . . . P j e j ≤ z < deg P 1 e 1 . . . P j e j P j+1 e j+1 .
We will consider the following cases:

Case 1: We have that
We note that We can now apply Corollary 6.4 to obtain where the second-to-last relation uses the fact that deg G ≤ (1 − α)y and z = α 10 y. Case 2: Suppose N satisfies case 2. Then, the associated P j+1 (from (23)) satisfies P j+1 e j+1 | N , deg P j+1 ≤ 1 2 z, and deg P j+1 e j+1 > 1 2 z. For a general prime P with deg P ≤ 1 2 z we denote e P ≥ 2 to be the smallest integer such that deg P e P > 1 2 z. We will need to note for later that Let us also note that for N with deg N ≤ deg X we have that where the last relation follows from the fact that z = α 10 y and deg G ≤ (1 − α)y. Case 3: Suppose N satisfies case 3. For the case where z ≤ q we have that w(z) = 1, meaning that the only possible value N could take is 1. At most this contributes O (1).
So, suppose that z > q, and so w(z) = log q z. Case 3 tells us that 1 2 z < deg B N ≤ z and Hence, as z −→ ∞, where the second-to-last relation follows from Lemma 6.7, and the last relation uses the fact that deg G ≤ (1 − α)y and z = α 10 y. Case 4: The case z < 1 is trivial, and so we proceed under the assumption that z ≥ 1. We have that .
and so where a = 2 20 αβ . So, continuing from (27), where X B is a monic polynomial of degree deg X −deg B such that deg X − B X B < y, and A B is a polynomial satisfying A B B ≡ A(mod G). Corollary 6.4 tells us that where the last relation follows from the fact that deg B ≤ z, z = α 10 y, and deg G ≤ (1 − α)y. Hence, continuing from (28): Finally, we wish to apply Lemma 6.8. This requires that r log q r ≤ z. Now, when 1 ≤ z ≤ q we have that w(z) = 1 and r 1 = z. Hence, r log q r ≤ z log q q = z.
When z > q we have that w(z) = log q z and r 1 = z w(z) . Hence, r log q r ≤ z log q z (log q z − log q log q z) ≤ z, since z > q. Hence, The proof now follows from (24), (25), (26), and (29).

Proof of Theorem 6.2
The proof of this theorem is almost identical to the proof of Theorem 6.1. Where we applied Corollary 6.4, we should instead apply Corollary 6.5. Also, the calculations respectively. Then, for all R ∈ A and j = 1, 2, 3, 4 we have that

Further preliminary results
Remark 7. 3 We must mention that, in the lemma and the proof, the implied constants may depend on j, for example; but because there are only finitely many cases of j that we are interested in, we can take the implied constants to be independent.
Proof First, we note that where We note further that For all R ∈ A and k = 0, 1, 2, 3 it is not difficult to deduce that g (k) The function (log x) k+1 x−1 is decreasing at large enough x, and the limit as x −→ ∞ is 0. Therefore, there exist an independent constant c ≥ 1 such that for k = 0, 1, 2, 3 and all A, Hence, taking n = ω(R), we see that where we have used the prime polynomial theorem and Lemma 4.4. So, by (30)-(33) and the fact that we deduce that Lemma 7.4 Let R ∈ A, and define z R := deg R − log q 9 ω(R) . We have that

Proof
Step 1: Let us define the function F for Re s > 1 by We can see that Now, let c be a positive real number, and define y R := q z R . On the one hand, we have that where the second equality follows from Lemma 7.1.
On the other hand, for all positive integers n define the following curves: Then, we have that Step 2: For the first integral in (35) If we apply the product rule for differentiation, then one of the terms will be Now we look at the remaining terms that arise from the product rule. By using the fact that ζ A (1+s) = 1 1−q −s and the Taylor series for q −s , we have for k = 0, 1, 2, 3, 4 that Similarly, By (38), (39), and Lemma 7.2, we see that the remaining terms are of order Hence, Step 2.2: Now we look at the remaining residue terms in (36 Step 3: We now look at the integrals over l 2 (n) and l 4 (n). There exists an absolute constant κ such that for all positive integers n and all s ∈ l 2 (n), l 4 (n) we have that F(s + 1)y R s ≤ κ|R| c+1 . One can now easily deduce for i = 2, 4 that Step 4: We now look at the integral over l 3 (n). For all positive integers n and all s ∈ l 3 (n) we have that We can now easily deduce that Step 5: By (34), (35), (42), (43) and (44), we deduce that Proof For s > 1 we define We can see that By comparing the coefficients of powers of q −s , we see that where the last relations follows from Lemma 7.5. We also note that This proves the first relation in the lemma. The second relation follows from Lemma 4.9.
Lemma 7.7 Let F, K ∈ M, x ≥ 0, and a ∈ F * q . Suppose also that 1 2 Proof We have that, H N ) where N , G , K are defined by H N = N , H G The third relation holds by Theorem 6.2 with β = 1 6 and α = 1 4 (one may wish to note that (K F, G ) = 1 and that the other conditions of the theorem are satisfied because The last relation follows from Lemma 4.10.
Lemma 7.8 Let F, K ∈ M and x ≥ 0 satisfy deg K F < x. Then, Proof The proof is similar to the proof of Lemma 7.7. We have that where N , G , K are defined by H N = N , H G where we define X := T x−deg H . We can now apply Theorem 6.1 to obtain that Lemma 7.9 Let F ∈ M and z 1 , z 2 be non-negative integers. Then, for all > 0 we have that where we define X (N ) = T z 1 +z 2 −deg N . We can now apply Theorem 6.1. One may wish to note that where 0 < α < 1 2 , as required. Hence, we have that Step 2.3: We now look at the sum By Lemma 7.8 we have that Proof The case where a = 1 is just Lemma 7.9. The proof of the case where a = 1 is very similar to the proof of Lemma 7.9. In fact it is easier, because the the case where deg AC = deg B D cannot exist: We would require that AC and B D are both monic, but also require that at least one of AC and B D have leading coefficient equal to a = 1.

Proposition 7.11
Let R ∈ M and define z R := deg R − log q 2 ω(R) . Also, let a ∈ F * q . Then, Proof We apply Lemma 7.10 with = 1 50 to deduce that So, where the second-to-last relation uses the following.

The fourth moment
We now proceed to prove Theorem 2.6. In the proof we implicitly state that some terms are of lower order than the main term and that is easy to check. We do not give the justification explicitly, although all the results one needs for a rigorous justification are given in Sect. 4.

Proof of Theorem 2.6
Let χ be a Dirichlet character of modulus R. By Lemmas 3.10 and 3.11, we have that We will show that * χ mod R |a(χ )| 2 has an asymptotic main term of higher order than * χ mod R |b(χ )| 2 and * χ mod R |c(χ )| 2 . From this and the Cauchy-Schwarz inequality, we deduce that * χ mod R |a(χ )| 2 gives the leading term in the asymptotic formula.
Step 1: We have that * where z R := deg R − log q 9 ω(R) . Let us look at the first term on the far-RHS of (47). We apply Lemma 4.12. When x = z R −deg N 2 and deg N ≤ z R , we have that 2 ω(R) x q x = O (1). Hence By (47) Step 1.2: For the second term on the far-RHS of (46) we simply apply Proposition 7.11. From this, Step 1.1, and (46), we deduce that * Step 2: We will now look at * For the second term we have that where we have used Lemma 7.9.Hence, Similarly, by using Lemma 7.10 for the even case, we can show, for a = 0, 1, 2, 3, that Hence, by using the Cauchy-Schwarz inequality, we can deduce that * χ mod R |c(χ )| 2 φ * (R) P prime P|R 1 − |P| −1 3 1 + |P| −1 (deg R) 3 ω(R).
Step 4: From steps 1 to 3, and the use of the Cauchy-Schwarz inequality (as described at the start of the proof), the result follows.