Moments of L-functions and Liouville-Green method

We show that the percentage of primitive forms of level one and weight $4k\rightarrow \infty$ for which the associated $L$-function at the central point is no less than $(\log{k})^{-2}$ is at least 20%. The key ingredients of our proof are the Kuznetsov convolution formula and the Liouville-Green method.


Introduction
Non-vanishing results for central values of L-functions in families have numerous applications, discovered, for example, in [6,13,17,19,33]. In particular, this paper is inspired by the work of Iwaniec and Sarnak [13], where they approached the problem of non-existence of Landau-Siegel zeros by studying the non-vanishing of automorphic Lfunctions at the critical point.
In the weight aspect they proved the following result. Let H 2k (1) be the space of primitive forms of level 1 and weight 2k ≥ 12, k ∈ N. For f ∈ H 2k (1), let L f (1/2) be the associated L-function at the critical point. For any ǫ > 0 one has (1.1) lim Moreover, the non-existence of Landau-Siegel zeros for Dirichlet Lfunctions of real primitive characters would follow if (1.1) is established with proportion strictly larger than 1/2. Inequality (1.1) was proved by Iwaniec and Sarnak [13] in 2000. Since then there were several attempts to obtain analogous result for individual weight. Independently Fomenko [7] and Lau & Tsang [21] showed that the proportion of non-vanishing without extra average over weight is at least 1/ log k. This is proved by computing the asymptotics of pure, unmollified first and second moments, and, therefore depends on k. Recently, Luo [23] showed that there is a strictly positive proportion of non-vanishing. However, his approach does not allow finding the exact proportion. The reason is that only an upper bound for the mollified second moment was proved in [23], while the full asymptotic expansion is required to make this result quantitive.
The aim of the present paper is to obtain the effective and strictly positive upper bound on the proportion of non-vanishing central Lvalues. Theorem 1.1. For any ǫ > 0 there exists k 0 = k 0 (ǫ) such that for any k ≥ k 0 and k ≡ 0 (mod 2) we have Our arguments rely in most part on the methods developed by Kuznetsov in 1990's. More precisely, we prove the exact formulas for moments of L-functions in the critical strip and express the error for the first moment in terms of the confluent hypergeometric function 1 F 1 (a, b; x) and the error for the second moment in terms of the Gauss hypergeometric function 2 F 1 (a, b, c; x). The exact statements are given by Theorems 3.2 and 4.2, respectively.
The most challenging problem is to optimize estimates of the error terms for the second moment that are given by two shifted convolution sums, namely where τ (n) is the number of divisors function and As the main tool we choose the Liouville-Green method (also called WKB approximation or the Liouville-Steklov method). It is one of the oldest approximation techniques widely applied, for example, in quantum mechanics. The idea of using it in analytic number theory belongs to Kuznetsov. The method is based on the observation that "close" differential equations have "close" solutions. Accordingly, in Section 5 we find differential equations satisfied by functions (1.3) and (1.4). The given equations can be approximated by other differential equations which have "simpler" functions as solutions. This allows approximating the error terms uniformly in k with any power of precision. See Theorems 5.13 and 5. 16.
The Liouville-Green approximation models the behavior of functions (1.3) and (1.4) using Y 0 and K 0 -Bessel functions with the large parameter k in the argument. This shows that in the required ranges Φ k (x) decays exponentially and φ k (x) is oscillatory. To smooth out the oscillations of φ k (x) one can average the error terms over weight with a suitable test function, reproving the result of Iwaniec and Sarnak (1.1), as shown in Theorem 8.8.
Asymptotic formulas for twisted moments have several other applications. For example, Hough [10] considered zero-density estimates for L-functions in the weight aspect. His proof is based on the asymptotic evaluation of the second moment near the critical line with the error term estimated as O(l 3/4 k −1/2+ǫ ) at the central point. The same error bound was obtained by Ng Ming Ho [27] using different approach. Our method (see Theorem 6.4) yields O(l 1/2 k −1/2+ǫ ).
Finally, techniques developed in the present paper can be beneficial in solving other problems in analytic number theory which involve analysis of special functions. In particular, our approach yields new results for moments of symmetric square L-functions in the weight aspect. See [2] for details.

Notations and technical lemmas
For v ∈ C let where ζ(s) is the Riemann zeta function. Let H 2k (1) be the set of primitive forms of level 1 and weight 2k ≥ 12. Every f ∈ H 2k (1) has a Fourier expansion of the form The Fourier coefficients of primitive forms are multiplicative For each f ∈ H 2k (1) the associated L-function is defined by and the associated symmetric square L-function is given by As a consequence of relation (2.6), for Reu > 1/2, ℜv = 0 we have Let Γ(s) be the Gamma function. The completed L-function satisfies the functional equation and can be analytically continued on the whole complex plane. It follows from equation (2.11) that L f (1/2) = 0 for odd k.
The harmonic weight is defined by where f, f 1 is the Petersson inner product on the space of level 1 holomorphic modular forms. Then the harmonic summation can be written as Consider two Bessel kernels For 3/2 > Rew > 2|ℜv| one has and for Rew > 2|ℜv| Consider the series Let x be a rational number x = d c with (d, c) = 1, c ≥ 1. Then the function D v (s, x) of two complex parameters s and v is meromorphic on the whole complex plane. If we fix v such that ℜv = 0 and v = 0, then D v (s, d/c), as a function of single variable s, has two simple poles at s = 1+v and s = 1−v with residues c −1−2v ζ(1+2v) and c −1+2v ζ(1−2v), respectively, and it is regular elsewhere. Also it satisfies the functional equation (see [24,Lemma 3.7 where dd * ≡ 1 (mod c) and γ(u, v) is defined by (2.17). For ℜs < 0 the following estimate is satisfied (see [24,Eq. 3.3.24]) Note that where {u} is a fractional part of u. For any ǫ > 0 we estimate the absolute value of (2.25), obtaining The Mellin transform of function f is defined by Proof. See, for example, [28, Section 3.1.3].

The first moment
In this section we derive the asymptotic formula for the first moment of automorphic L-functions at the critical point.
This theorem is a consequence of more general statement. Let us define two auxiliary functions where nn * ≡ 1 (mod c).
Furthermore, for t ∈ R, T = 1 + |t| Proof. Our arguments follow the proof in [1]. The only difference is the second main term which appears due to an additional pole when level is equal to 1.
Originally, Lemma 4.1 was proved by Kuznetsov in his doctoral thesis (1981) and also published in [20]. Unfortunately, book [20] is hard to find, so we provide all details here.
The proof of Lemma 4.1 is based on the properties of the series D v (s, x) defined by equation (2.20). By the inverse Mellin transform The change of order of integration and summation in the formula above is allowed since (b) |φ(2s)|ds < ∞ and the series D v (s, d/c) converges absolutely for ℜs = b > 1.
Moving the contour of integration to ℜs = δ with σ 0 < δ < 0, we cross two simple poles of D v s, d c at the points 1 + v and 1 − v. Computation of the residues gives the first two summands on the righthand side of (4.1). To justify this contour shift, we show that for
Finally, we compute Now we can move the contour of integration to ℜs = α such that 3/4 < α < 1. Then the result follows from Lemmas 2.3 and 2.2, as we now show. Let Parameter α is chosen such that condition (2.28) is satisfied for g 1 (x), i.e.
This concludes the proof of Lemma 4.1.

4.2.
Convolution formula for the second moment. Exact formulas for moments is a useful tool for understanding the structure of mean values and obtaining asymptotic expansions. Here we use the version by Kuznetsov. Similar formula was also independently obtained by Iwaniec and Sarnak [13].
The summand E(l; u, v) can be expressed in terms of hypergeometric functions We multiply both sides of the Petersson trace formula (2.14) by and sum over n ≥ 1. Using relation (2.9), we obtain the first summand on the right-hand side of (4.5) plus the non-diagonal contribution Applying Weil's bound for Kloosterman sums and standard estimates for J-Bessel function, the double series can be estimated as follows Thus for ℜu > 3/4 the series is absolutely convergent and we can change the order of summation in M N D . Replacing Kloosterman sum by its definition in the inner series and applying Lemma 4.1 with we obtain Ramanujan's identity (2.4) and [8, Eq. 6.561 (14)] for −1/4 < ℜu < k allow expressing the first two terms in M N D as the third and fourth summands on the right-hand side of (4.5). The second summand in (4.5) comes from the third term in M N D when n = l by applying [8, Eq. 6.574(2)] for 0 < ℜu < k. Consider If n < l we apply identity [8, Eq. 6.574] for −1/2 < ℜu < k to evaluate the integral over x and Ramanujan's identity (2.4) to compute the sum over c, obtaining the first term in (4.6). Analogously, for n > l we obtain the second term in (4.6) using [8, Eq. 6.574 (1)] . Finally, the third term in (4.6) comes from by applying [8, Eq. 6.576(3)] for ℜu < k and identity (2.4). Note that the left-hand side of convolution formula (4.5) is entire function of u and v. Since the right-hand side of (4.5) is regular function for |ℜv| + |ℜu| < k − 1.

The Liouville-Green method
Our main references are chapter 12 of book [25] and paper [5]. We assume that k is an even positive integer. 5.1. Some properties of φ k . In this section we study the function where φ k (x; u, v) is defined by equation (4.7). Letting u = 0 and computing the limit as v → 0 by L'Hospital's rule, we obtain .
Differentiation with respect to v gives .
Computing the limit as u → 0 by L'Hospital rule, we have Letting v = 0, we have .
Lemma 5.3. The following series representation holds Proof. By equation (5.1) and Euler's reflection formula Then the assertion follows by applying [26,Eq. 15.2.4].
Lemma 5.4. The function φ k (x) satisfies the differential equation where Coefficients A(n), B(n), C(n) are defined by

They satisfy recurrence relations
Let us denote Using the recurrence relations above, we compute D(α 1 ), D(α 2 ), D(α 3 ) and prove the Lemma by showing that Lemma 5.5. Let y = y(x) be a solution of differential equation Then z(x) = y(x)/α(x) satisfies equation is a solution of differential equation Lemma 5.7. Assume that k is an even positive integer. Then Proof. Using formula [26,Eq. 15.8.25] and Euler's reflection formula, we have for some complex variable a. Setting a = k we obtain Equation (5.11) follows by taking x = 1/2. Equation (5.12) is obtained by differentiation of 2 F 1 (k, 1 − k, 1; x) with respect to x using the series representation This representation is a consequence of [26, Eq. 15.2.4] for even positive integer k.
Lemma 5.8. For even positive integer k one has Proof. By formulas (5.2) and (5.11) we have .
Differentiating G 1 and G 2 with respect to ǫ at the point ǫ = 0 and summing the results, we have Setting z = 1/2 we prove the Lemma.

5.2.
Asymptotic approximation of φ k . We apply the Liouville-Green method to find a uniform approximation of the function φ k (x). In Corollary 5.6 we showed that y(x) = x(1 − x)φ k (x) is a solution of differential equation (5.10). This equation is a particular type of [5, Eq. 1.1] when α = 0. Let (5.14) u , .
Then equation (5.10) can be written as Note that x 2 g(x) → −1/4 as x → 0. We would like to transform equation (5.16) into the following shape In order to obtain equation (5.17) we make the coefficient before dZ dξ vanish by requiring Next, we assume that −α 4 (x)f (x) = 1/(4ξ). This implies .
Lemma 5.11. For λ 1 = 1 16 + 405 32π 2 one has Proof. Note that with our choice of λ 1 the value of c 1 is non-zero. Therefore, by (5.42), (5.47), (5.50), (5.51) we have By Lemma 5.10 We choose λ 1 such that the first summand in the formula above is zero, so that Combining all results we find that C J = O(u −5 ) = O(k −5 ).

5.3.
Asymptotic approximation of ψ k and Φ k . In this section we estimate the functions defined by (4.9) and (4.10), namely By [4, Eq. 1-3, p. 105] Hence (5.60) ψ k l n = Φ k l n + l and it is sufficient to consider only function Φ k .
Let u := k − 1/2 and Lemma 5.14. The function y(x) = 2 F 1 (k, k, 2k; x)x k √ 1 − x is a solution of differential equation Proof. The hypergeometric function F (x) = 2 F 1 (k, k, 2k; x) satisfies equation Applying Lemma 5.5 with The assertion follows by rearranging the expression in brackets.
Note that equation (5.62) differs from (5.16) by the sign in f (x). According to [25,Chapter 12] this means that one has to choose I and K Bessel functions (instead of Y and J) as approximation functions. Consequently, we transform (5.62) to the type This can be done similarly to the previous case by making the change Note that as ξ → 0 the function ψ(ξ) is analytic. Removing the term with ψ(ξ)/ξ in (5.63) we obtain where L 0 is either K 0 or I 0 Bessel function. In general, Therefore, according to [25, Eq. 2.09, Chapter 12] the solution of differential equation (5.63) can be found in the form To determine coefficients A(n; ξ) and B(n; ξ) we introduce two functions Furthermore, using formulas 10.29.2 and 10.29.3 we prove that Substituting equation (5.70) into (5.63) one has where C(n; ξ) = A ′′ (n; ξ) + A ′ (n; ξ) ξ − ψ(ξ) ξ A(n; ξ) + B ′ (n; ξ) + B(n; ξ) 2ξ , Setting C(n; ξ) = D(n; ξ) = 0 we find recurrence relations Note that The expression for B(1; ξ) is quite complicated, so we do not write it explicitly. The only thing we need to show is that is bounded. Using recurrence relation (5.76), we find (1; x)) . Therefore, as required. Note that Hence for n > 1 the variation Theorem 5.15. For each value of u and each nonnegative integer N equation (5.63) has solution Z K (ξ) which is infinitely differentiable in ξ on interval (0, ∞) and is given by In particular, for N = 1 The solution of differential equation (5.63) is recessive as ξ → ∞. Another recessive solution is Z K (ξ) defined by (5.70) with L v = e πiv K v . Therefore, there exists C K = C K (u) such that Computing the limit as ξ → ∞ of the left and right-hand sides of equation (5.89), we find Since we have (5.93) To sum up, we proved the following result.

Error terms for individual weight
Lemma 6.1. One has Proof. The assertion follows from Theorem 4.2 by computing the limit as u → 0, v → 0. The error terms can be simplified as follows. First, by equation (5.3) Second, by equation (5.60) Therefore, Making the change of variables ξ = x 2 and splitting the integral into two parts Now we estimate the second term Then by equation (5.89) Note that ξ 0 ∼ l −1 . Then for l ≪ k 2 one has Finally, for some constant c > 0 independent of k Proof. We apply equation (5.42) with n/l =: sin 2 Using property (5.3) we find Thus n/l ≤ 1/2 and ξ ≤ π 2 /4. Since n/l ≥ 1/l one has ξ ≫ 1/l. Hence Applying Theorem 5.9 with N = 0, we find Finally, Combining all results together we prove the asymptotic formula.

Error terms on average
In this section we estimate the error terms averaged over k with a test function h ∈ C ∞ 0 (R + ), which is non-negative, compactly supported on interval [θ 1 , θ 2 ] such that θ 2 > θ 1 > 0 and (7.1) h (n) 1 ≪ 1 for all n ≥ 0. Let We denote the averaged moments as follows Lemma 7.1. One has Proof. We average over k the result of Lemma 6.1. To compute the main term, we use [26,Eq. 5.11.2], namely where B 2r are the Bernoulli numbers. Note that By Poisson's summation formula one has If n = 0 this is equal to If n = 0 we integrate by parts a ≥ 2 times and estimate the expression by its absolute value, obtaining Similarly, Lemma 7.2. For l ≪ K 2−ǫ , for any A > 0 Contribution of φ 2k (1/2)τ 2 (l/2) can be estimated using formula (5.13) Applying formula (5.42) we have φ 2k n l = C Y Z Y (4 arcsin 2 n/l) + C J Z J (4 arcsin 2 n/l) (2 arcsin n/l) 1/2 (2n/l) 1/4 (1 − n/l) 1/4 .
Let u := 2k − 1/2, ξ := 4 arcsin 2 n/l. Then by equations (5.39), (5.55) we obtain Therefore, contribution of the term with C J Z J is bounded by Now we estimate contribution of the term with C Y Z Y , namely We use the series representation (5.57) for C Y with sufficiently large n. The main contribution comes from the first summand −2π.
The Y -Bessel functions have oscillatory behavior. According to equation [8, Eq. 8.451(2)] one has [8,Eq. 8.451 (7,8)]. Since u √ ξ > 1 the only difference between Y 0 (u √ ξ) and Y 1 (u √ ξ) is the shift on π/2 in the oscillating multiples. Thus one can consider only Y 0 (u √ ξ). Contribution of R 1 , R 2 is majorized by It is sufficient to estimate Let g(y) := 1 4 Ky(−2πm±4 arcsin n/l), then writing the sine in terms of exponentials we have Integration by parts a ≥ 2 times yields Note that 0 < arcsin n/l ≤ π/4. If m = 0 one has Consequently, Finally, we obtain the main result of this section.
Theorem 7.4. For any ǫ > 0, any a ≥ 2, l ≪ K 2 one has The asymptotic formula for the averaged first moment follows from Theorem 3.1 .
Note that x m ≪ 1, and, therefore, Averaging over k we prove the following estimate.
Note that on the new contour ζ(s + 1) has no zeros (see, for example, [34,Theorem 4,p. 33]) and the integral is bounded by Using the saddle point method, we estimate the last integral as Finally, the residue at s = 0 is equal to Lemma 8.4. Let M = K ∆ . For any ǫ > 0 there is K 0 = K 0 (ǫ) such that for every K ≥ K 0 one has for any ∆ < 1 − ǫ.
The contribution of the first three integrals above is negligible and can be estimated similarly to the proof of Lemma 8.3. The fourth integral can be bounded by a constant and, therefore, its contribution to P 1 is O((log M) −4 ). The main term is given by the residue at s 1 = s 2 = z = 0. The function f (s 1 , s 2 , z) has a simple pole at s 2 = 0. Hence res s 1 =s 2 =z=0 f (s 1 , s 2 , z) = 2πi res s 1 =z=0 k z M s 1 ζ(s 1 + 1)α(s 1 , 0, z) z 2 s 3 1 ζ 2 (s 1 + z + 1) .
Now the key idea is to replace L(sym 2 f, 1) by a short Dirichlet polynomial Repeating the arguments of [17] and using [ |α f | ≪ (log k) A , for some A > 0,