Ratios conjecture for quadratic Hecke $L$-functions in the Gaussian field

We develope the $L$-functions ratios conjecture with one shift in the numerator and denominator in certain ranges for the family of quadratic Hecke $L$-functions in the Gaussian field using multiple Dirichlet series under the generalized Riemann hypothesis. We also obtain an asymptotical formula for the first moment of central values of the same family of $L$-functions, obtaining an error term of size $O(X^{1/2+\varepsilon})$.


Introduction
The L-functions ratios conjecture formulated in [6, Section 5] by J. B. Conrey, D. W. Farmer and M. R. Zirnbauer gives a general recipe to predict both the main and lower-order terms in the asymptotic formulas for the sum of ratios of products of shifted L-functions.This conjecture has been applied to study a wide variety of important problems such as the density conjecture of N.Katz and P. Sarnak [19,20] on the distribution of zeros near the central point of a family of L-functions, the mollified moments of L-functions, the discrete moments of the Riemann zeta function and its derivatives.A detailed description of these applications can be found in [7].Some results can be found in the literature concerning the ratios conjecture.For certain ranges of parameters, the ratios conjecture for quadratic L-functions was established by H. M. Bui, A. Florea and J. P. Keating [4] over function fields and by M. Čech [5] over Q by further assuming the generalized Riemann hypothesis (GRH).
The results of Čech [5] were obtained by utilizing the method of double Dirichlet series, a powerful tool that has been previous deployed to investigate related issues such as moments of central values of families of L-functions.In this paper, we further apply this approach to study the ratios conjecture for quadratic Hecke L-functions over the Gaussian field.To state our result, we write K = Q(i) for the Gaussian field and O K = Z[i] for its ring of integers throughout the paper.Further N (n) denotes the norm of any n ∈ K and ζ K (s) the Dedekind zeta function of K.It is shown in Section 2.1 below that every ideal in O K co-prime to 2 has a unique generator congruent to 1 modulo (1+i) 3 which is called primary.Let χ m be the quadratic symbol m • defined in Section 2.1, which can be viewed as an analogue in K to the Kronecker symbol.As χ m equals 1 on the group of units of K, we may regard it as a quadratic Hecke character of trivial infinite type and denote the associated L-function by L(s, χ m ).Furthermore, we use the notation L (c) (s, χ m ) for the Euler product defining L(s, χ m ) but omitting those primes dividing c.
We first establish a result concerning the ratios conjecture with one shift in the numerator and denominator for the family of Hecke L-functions averaged over all quadratic Hecke characters.
We shall establish Theorem 1.1 following the line of approach in the proof of [5,Theorem 1.2].A key ingredient involved is a functional equation of L-functions attached to the general quadratic Hecke character given in Proposition 2.5.
As an application of Theorem 1.1, we note that the case β → ∞ of (1.1) leads to a result concerning the smoothed first moment of values of quadratic Hecke L-functions except for a large error term.We shall in fact use the method in this paper to achieve the following valid asymptotic formula in Section 4 unconditionally.Theorem 1.2.Let w(t) be the same as in Theorem 1.1.For 1/2 > ℜ(α) > 0, we have, for any ε > 0, We note here that the main term in (1.4) are identified with exactly the ones obtained by taking β → ∞ in the main term of (1.1).As the error term in (1.4) is uniform for α, we can further take the limit α → 0 + and deduce the following asymptotic formula for the smoothed first moment of central values of quadratic Hecke L-functions.
Corollary 1.3.Let w(t) the same as in Theorem 1.1.We have, for any ε > 0, where Q is a linear polynomial whose coefficients depend only on the absolute constants and w(1) and w ′ (1).
We omit the explicit expression of Q here as our main focus here is the error term.One can compute it by working with the function A(s, w) defined in (4.2) which has a double pole at s = 1, w = 1/2.Evaluation of the first moment of central values of the classical quadratic family of Dirichlet L-functions goes back to M. Jutila [18].The error terms in Jutila's results were subsequently improved in [15,22,23].A result of D. Goldfeld and J. Hoffstein in [15] implies that the error term is of size O(X 1/2+ε ) for a smoothed first moment, a result which was later obtained by M. P. Young in [23] using a recursive approach.We point out here that a result similar to Corollary 1.3 has been achieved by the first-named author [11] using Young's method.
In [9], A. Diaconu, D. Goldfeld and J. Hoffstein applied the method of multiple Dirichlet series to obtain an asymptotical formula for the third moment of quadratic twists of Dirichlet L-functions.It is also pointed out by them that there were many advantages to treat multiple Dirichlet series as functions of several complex variables.The general philosophy is then to develop enough functional equations for a corresponding multiple Dirichlet series to obtain meromorphic continuations of the series to a region as large as possible.A typical way of obtaining the desired functional equations is to make use of the available functional equations for L-functions associated to primitive characters and then to modify the multiple Dirichlet series involved appropriately so that such functional equations hold for L-functions associated to non-primitive characters as well.See [8] and [10] for further applications of this approach in both function fields and number fields setting.
In comparison, Čech obtained functional equations for the multiple Dirichlet series studied in [5] for all characters by modifying the usual approach to acquire such functional equations for L-functions attached to primitive characters.This process involves with a type of Poisson summation formula developed by K. Soundararajan in [21] when studying the non-vanishing of the central values of quadratic Dirichlet L-functions.The advantage of Čech's method is that one no longer needs to modify the multiple Dirichlet series involved to obtain the desired functional equations.Our result in this paper can be viewed as another application of the approach of Čech by adopting both multiple Dirichlet series and harmonic analysis in the setting of quadratic twists of Hecke L-functions in the Gaussian field.
Finally, we remark here that a result similar to Corollary 1.3 was obtained in [13,Theorem 1.1], also utilizing double Dirichlet series.Our approach here in establishing Theorem 1.2 (and thus Corollary 1.3) is however different and hence represents another proof of such a result.

Preliminaries
We first include some auxiliary results needed in the paper.
2.1.Quadratic Hecke characters and quadratic Gauss sums.Recall that K = Q(i) and it is well-known that K has class number one.Set U K = {±1, ±i} and D K = −4 for the group of units in O K and the discriminant of K, respectively.We reserve the letter ̟ for a prime in O K by which we mean that (̟) is a prime ideal.We say an element n ∈ O K is odd if (n, 2) = 1 and an element n ∈ O K is square-free if no prime square divides n.Note that n is square-free if and only if µ [i] (n) = 0, where µ [i] is the Möbius function on O K .Notice that (1 + i) is the only prime ideal in O K that lies above the integral ideal (2) in Z.
* denote the group of reduced residue classes modulo q which is the multiplicative group of invertible elements of O K /(q).Note that O K /((1 + i) 3 ) * is isomorphic to the cyclic group of order four generated by i.This implies that every ideal in O K co-prime to 2 has a unique generator congruent to 1 modulo (1 + i) 3 .This generator is called primary.Furthermore, (O K /(4)) * is isomorphic to the direct product of a cyclic group of order 4 and a cyclic group of order 2.More precisely, we have Note that the class 1 (mod (1 + i) 3 ) gives rise to two classes 1 + 2(1 + i)k, k ∈ {0, 1} modulo 4. It follows from this that an element n = a + bi ∈ O K with a, b ∈ Z is primary if and only if a ≡ 1 (mod 4), b ≡ 0 (mod 4) or a ≡ 3 (mod 4), b ≡ 2 (mod 4).This result can be found in Lemma 6 on [16, p. 121].We shall refer to these two cases above as n is of type 1 and type 2, respectively.
Let 0 = q ∈ O K and χ : (O K /(q)) * → S 1 := {z ∈ C : |z| = 1} be a homomorphism.Following the nomenclature of [17, Section 3.8], we refer χ as a Dirichlet character modulo q.We say that such a Dirichlet character χ modulo q is primitive if it does not factor through (O K /(q ′ )) * for any divisor q ′ of q with N (q ′ ) < N (q).For any odd n ∈ O K , the symbol • n stands for the quadratic residue symbol modulo n in K.For an odd prime ̟ ∈ O K , the quadratic symbol is defined for a ∈ O K , (a, ̟) = 1 by a ̟ ≡ a (N (̟)−1)/2 (mod ̟), with a ̟ ∈ {±1}.If ̟|a, we define a ̟ = 0. Then the quadratic symbol is extended to any odd composite n multiplicatively.We further define The following quadratic reciprocity law (see [12, (2.1)]) holds for two co-prime primary elements m, n Moreover, we deduce from Lemma 8.2.1 and Theorem 8.2.4 in [1] that the following supplementary laws hold for primary n = a + bi with a, b ∈ Z: Note that the quadratic symbol • n defined above is a Dirichlet character modulo n.It follows from (2.2) that the quadratic symbol ψ i := i • defines a primitive Dirichlet character modulo 4. Also, (2.2) implies that the quadratic symbol ψ 1+i := 1+i • defines a primitive Dirichlet character modulo (1 + i) 5 .This can be inferred by noting that As 5 ≡ 1 (mod 4) and ψ 1+i (5) = −1, we get that ψ 1+i must be primitive.
Furthermore, we observe that there exists a primitive quadratic Dirichlet character ψ 2 modulo 2 since (O K /(2)) * is isomorphic to the cyclic group of order two generated by i.Then we have ψ 2 (n) = −1 for n ≡ i (mod 2).
For any l ∈ Z with 4|l, we define a unitary character χ ∞ from C * to S 1 by: The integer l is called the frequency of χ ∞ .Now, for a given Dirichlet character χ modulo q and a unitary character χ ∞ , we can define a Hecke character ψ modulo q (see [17,Section 3.8]) on the group of fractional ideals I K in K, such that for any (α If l = 0, we say that ψ is a Hecke character modulo q of trivial infinite type.In this case, we may regard ψ as defined on O K instead of on I K , setting ψ(α) = ψ((α)) for any α ∈ O K .We may also write χ for ψ as well, since we have ψ(α) = χ(α) for any α ∈ O K .We then say such a Hecke character χ is primitive modulo q if χ is a primitive Dirichlet character.Likewise, we say that χ is induced by a primitive Hecke character We now define an abelian group CG such that it is generated by three primitive quadratic Hecke characters of trivial infinite type with corresponding moduli dividing (1 + i) 5 .More precisely, and the commutative binary operation on CG is given by As we shall only evaluate the related characters at primary elements in O K , the definition of such a product is therefore justified.
We note that the product of χ c for any primary c with any ψ j ∈ CG gives rise to a Hecke character of trivial infinite type.To determine the primitive Hecke character that induces such a product, we observe that every primary c can be written uniquely as c = c 1 c 2 , c 1 , c 2 primary and c 1 square-free.
The above decomposition allows us to conclude that if c 1 is of type 1, then χ c • ψ j for j ∈ {1, i, (1 + i) 5 c 1 for j = 1, i, 1 + i and i(1 + i), respectively.We make the convention that for any primary n ∈ O K , we shall use χ n •ψ j for any ψ j ∈ CG to denote the corresponding primitive Hecke character χ that induces it throughout the paper.
For any complex number z, we define With any r ∈ O K , the quadratic Gauss sum g(r, χ) associated to any quadratic Dirichlet character χ modulo q is defined by For the special case when χ = • n of any primary n, we further define x n e rx n .
Let ϕ [i] (n) be the number of elements in (O K /(n)) * .We recall from [12, Lemma 2.2] the following explicitly evaluations of g(r, n) for primary n.
(ii) Let ̟ be a primary prime in O K .Suppose ̟ h is the largest power of ̟ dividing k.
For any primary c ∈ O K , the Chinese remainder theorem implies that x = 2y + cz varies over the residue class modulo 2c as y and z vary over the residue class modulo c and 2, respectively.From this, we infer that for j = 1 and 2, Observe that we have = i c and that ψ j (c) = 1 since we have c ≡ 1 (mod 2).Also, using that the representation of (O K /(2)) * can be chosen to consist of 1, i, we see via a direct computation that g(r, ψ j ) = 3. Functional equations of L(s, χ).Let χ be a primitive Hecke character of trivial infinite type modulo q.A well-known result of E. Hecke shows that L(s, χ) has an analytic continuation to the whole complex plane and satisfies the functional equation (see [17,Theorem 3.8 Note that |W (χ)| = 1 and that D K = −4 in our situation.We then conclude from (2.4) and(2.5)that For our purpose here, we need to generalize the above functional equation to non-primitive quadratic characters.To that end, we first recall that the Mellin transform f of any function f is defined to be The following Poisson summation formula for smoothed character sums over all elements in O K is [12, Lemma 2.7].
Lemma 2.4.Let χ be a Dirichlet character modulo n.For any smooth function W : R + → R, we have for X > 0, where Applying the above lemma with W (s) = e −s , we first compute Now it follows from this and (2.8) that we have We apply the above identity to the Dirichlet character χ n modulo 2n defined by χ n = ψ j • • n when n is of type j for j = 1, 2. We note that χ n (0) = 0 and by Lemma 2.2 that g(0, n) = 0 unless n = , where we write for a perfect square in O K .We thus conclude for n = , We consider the Mellin transform of the left-hand side above.This leads to (2.10) where the last equality above follows by noting that χ n (c) = 1 for c ∈ U K and hence can be regarded as a Hecke character modulo 2n of trivial infinite type.
On the other hand, the Mellin transform of the right-hand side of (2.9) equals The change of variable u = N (k)/(4yN (2n)) transforms the last integral above to We deduce from (2.10)-(2.12) that 4(2π The following result summarizes our discussions above.
Proposition 2.5.For any primary n ∈ O K , n = , we have 2.6.A mean value estimate for quadratic Hecke L-functions.In the proof of Theorem 1.1, we need the following lemma, a consequence of [2, Corollary 1.4], which gives an upper bound for the second moment of quadratic Hecke L-functions.
Lemma 2.7.Suppose s is a complex number with ℜ(s) ≥ 1  2 and that |s − 1| > ε for any ε > 0. Then we have * where * henceforth denotes the sum over square-free elements in O K .
2.8.Some results on multivariable complex functions.We include in this section some results from multivariable complex analysis.First we need the notation of a tube domain.
For a set U ⊂ R n , we define T (U ) = U + iR n ⊂ C n .We have the following Bochner's Tube Theorem [3].The Mellin inversion renders that (3.1) A s, Here the last equality above follows from the quadratic reciprocity law given in (2.1).
To deal with the last integral in (3.1), we need to understand the analytical properties of A(s, w, z).We shall do so in the following sections.
3.1.First region of absolute convergence of A(s, w, z).We start with the series representation for A(s, w, z) given by the first equality in (3.2) to see that Here ω [i] (h) denotes the number of distinct primes in O K dividing h and the last estimate above follows from the well-known bound We therefore deduce that We treat the right-hand side expression above by noting the following bound from [17,Theorem 5.19] that asserts on GRH, Moreover, we deduce from (2.14) and Cauchy's inequality that for any ℜ(s) ≥ 1/2 and that |s It follows from the above that except for a simple pole at w = 1, both sums of the right-hand side expression in (3.4) are convergent for ℜ(s) > 1, ℜ(w) ≥ 1/2, ℜ(z) > 1/2.
On the other hand, we apply the functional equation given in (2.6) for L(w, χ n ) when ℜ(w) < 1/2 to see that both sums of the right-hand side of (3.4) converge for ℜ(s + w) > 3/2, ℜ(w) < 1/2, ℜ(z) > 1/2.We thus conclude that the function A(s, w, z) converges absolutely in the region We also deduce from the last expression of (3.2) that A(s, w, z) is given by the series As noted before, χ n is a Hecke character modulo 2n of trivial infinite type.This information will be needed later in the application of the convexity bound and the evaluation of the relevant Gauss sums using Lemma 2.2.We write mk = (mk) 0 (mk) 2  1 with (mk) 0 primary and square-free.The bound in (3.3) renders Now we use the convexity bound for L(s, χ (mk)0 ) (see [17, Exercise 3, p. 100]) which asserts that Applying the above estimations for the case ℜ(s) ≥ 1/2, together with the functional equation in (2.6) for L (s, χ mk ) with ℜ(s) < 1/2 to (3.7), gives that, except for a simple pole at s = 1 arising from the summands with mk = , the function A(s, w, z) converges absolutely in the region Thus by our discussions above and Theorem 2.10, (s − 1)(w − 1)A(s, w, z) converges has a holomorphic continuation to the region S 2 .

3.2.
Residue of A(s, w, z) at s = 1.It follows from (3.7) that A(s, w, z) has a pole at s = 1 arising from the terms with mk = .In this case, we have Recall that the residue of ζ K (s) at s = 1 equals π/4.Then we have where we denote by a t (n) for any t ∈ C the multiplicative function such that a t (̟ k ) = 1 − 1/(N (̟) t ) for any prime ̟.We express the last sum as an Euler product and obtain, via a direct computation similar to that done in [5, (6.8)], that, for w = 1/2 + α and z = 1/2 + β, . (3.11) 3.3.Second region of absolute convergence of A(s, w, z).Applying (3.7), together with the functional equation from Proposition 2.5, gives that (3.12) We observe first that, upon setting mk = l, A 1 (s, w, z) =ζ (2) It follows from the above that except for a simple pole at s = 1, A 1 (s, w, z) is holomorphic in the region Next, we apply the functional equation (2.13) for L (s, χ mk ) in the case mk = .This gives where C(s, w, z) is given by the triple Dirichlet series By (3.10), (3.14) and the functional equation (3.15), we see that C(s, w, z) is initially defined in the region To extend this region, we exchange the summations in C(s, w, z) and setting mk = l to obtain that We now apply (2.3) to evaluate g (q, χ l ) and apply (2.2) to detect whether l is of type 1 or not using the character sums 1  2 (χ 1 (l) ± χ i (l)).This allows us to recast C 1 (s, w, z) and C 2 (s, w, z) as where D(w, t, q; χ) := l primary χ(l)g (q, l) a t (l) N (l) w , D 1 (w, t, q) := l primary g q, l 2 a t (l) Here a t is the function defined earlier.
We have the following result concerning the analytic properties of D(w, t, q; χ) and D 1 (w, t, q).Lemma 3.4.The functions D(w, t, q; χ 1 ), D(w, t, q; χ i ) and D 1 (w, t, q) have meromorphic continuations to the region except for a simple pole at w = 3/2 when q = i and t = 0 for D(w, t, q; χ 1 ) and a simple pole at w = 3/2 when q = and t = 0 for D(w, t, q; χ i ).Moreover, for ℜ(w) > 1 + ε and ℜ(t + w) > 1 + ε, we have, away from the possible pole, Proof.As the situations are similar, we consider only the case D(w, t, q; χ i ) here.We recast it as where P g denotes the product over generic odd primes not dividing q and P n denotes the rest.Lemma 2.2 gives The last Euler product is absolutely convergent and ≪ 1 for This completes the proof of the first assertion of the lemma.The second assertion of the lemma here can be proved in a manner similar to [5, Lemma 6.1 (2)].
The above lemma now allows us to extend C(s, w, z) to the region Using (3.12)-(3.15)and the above, we can extend (s − 1)(w − 1)(s + w − 3/2)A(s, w, z) to the region Note that the condition ℜ(s + 2w) > 1 is superseded by ℜ(2w) > 1 and ℜ(s + w) > 1 so that we have Note that the point (1, 1/2, 1/2) is in the closure of S 2 and the point (0, 1, 1) is in the closure of S 4 .We then get that the convex hull of S 2 and S 4 contains Applying Theorem 2.10 yields that (s − 1)(w − 1)(s + w − 3/2)A(s, w, z) can be holomorphically continued to the region S 5 .
Also, if q = , we have by Lemma 2.2 and a direct computation (similar to that in [5, (6.46)]) that We conclude from the above by another direct computation (with the help of the identity given in [5, (6.51)]) that Res w=3/2 D(w, z − w, q = ; χ i ) = π 4 where P (z) is given in (1.2).
We now apply (3.15) and (3.16) to conclude that Setting w = 1/2 + α and z = 1/2 + β, Note that the functional equation (2.6) implies The above allows us to recast the expression in (3.21) as 3.6.Bounding A(s, w, z) in vertical strips.In this section, we give estimations of |A(s, w, z)| in vertical strips, which is needed in order to evaluate the integral in (3.1).
Using Proposition 2.11, we obtain that in the convex hulls of S2 , S0 and S1 , we have Moreover, by (3.13), the convexity bound given in (3.9) for ζ K (s), the functional equation given in (2.6) for ζ K (s) for ℜ(s) < 1/2 and (2.7) that in the region S3 , we have We encounter two simple poles at s = 1 and s = 1 − α in this process, with the corresponding residues being given in (3.11) and (3.23).These give the main terms in (1.1).
To estimate the integral on the new line, we use (3.29) and obtain that on this line Moreover, note that integration by parts implies that for any integer E ≥ 0, We apply this with (3.30) to get that the integral on the new line is bounded by the error term given in (1.1).This completes the proof of Theorem 1.1.
Proof.As the situations are similar, we give the details only for the case D(w, q; χ i ) here.Recasting it as the first assertion of the lemma readily follows.
We further note that when ℜ(w) > 1, Applying the above together with the estimations given in (3.9) to bound L (w − 1/2, χ q ), we see that the second assertion of the lemma follows.This completes the proof.

4.2.
Residue of A(s, w) at s = 1.We see from (4.4) that A(s, w) has a pole at s = 1 coming from the summands with m = .In this case, we have Res s=1 A(s, w) = π 8 m= m primary a 1 (m) N (m) w , where we recall that a 1 (n) denotes the multiplicative function with a 1 (̟ k ) = 1 − 1/N (̟).

Conclusion.
We then shift the integral in (4.1) to ℜ(s) = 1/2 + ε to encounter two simple poles at s = 1 and s = 1 − α in the process, with the corresponding residues given in (4.11) and (4.13).These give the main terms in (1.4).We then apply (3.31) together with the bound given in (4.16) to infer that the integral on the new line is bounded by the error term given in (1.4).This completes the proof of Theorem 1.2.

Theorem 2 . 10 . 3 .
Let U ⊂ R n be a connected open set and f (z) be a function that is holomorphic on T (U ).Then f (z) has a holomorphic continuation to the convex hull of T (U ).The convex hull of an open set T ⊂ C n is denoted by T .Then we quote the result from [5, Proposition C.5] concerning the modulus of holomorphic continuations of functions in multiple variables.Proposition 2.11.Assume that T ⊂ C n is a tube domain, g, h : T → C are holomorphic functions, and let g, h be their holomorphic continuations to T .If |g(z)| ≤ |h(z)| for all z ∈ T , and h(z) is nonzero in T , then also |g(z)| ≤ | h(z)| for all z ∈ T .Proof of Theorem 1.1