| Estimates of the values of L-functions at the central point s=1/2 are subject of intensive studies in various aspects. The positivity of the Dirichlet L-functions with real characters at s=1/2 would yield quite remarkable effective lower bounds for the class number of imaginary quadratic fields. A good positive lower bound for the central values of Hecek L-functions would rule out the existence of the Landau-Siegel zero. Moreover, the nonvanishing of certain Rankin-Selberg L-functions is a crucial ingredient in the current development of the generalized Ramanujan conjecture [32].In 2005, Ramakrishnan and Rogawski [34] derived a simultaneous nonvanishing result for the central values of the product of two L-functions, using the relative trace formula. They proved that there are infinitely many N>0, such that L(1/2,φ×χ)L(1/2,φ)≠0, whereφis a newform of weight k for the congruence subgroupΓo(N) andχa Dirichlet character. There are a number of other stronger nonvanishing results in terms of percentage of nonvanishing for a single L-function, see [1], [2], [18], [23], [24], [25], [31], [35], for example. Recently, Li [29] considered the simultaneous nonvanishing problem of products of Rankin-Selberg L-functions on GL(3) x GL(2) and Maass L-functions on GL(2) at 1/2. She proved that there are infinitely many such L-functions such that they are nonvanishing simultaneously at 1/2.In this paper, we consider the sum of products of Rankin-Selberg L-functions on GL(2) x GL(2) and Maass L-functions on GL(2) at 1/2. To begin with, let {uj} be an orthonormal basis of even Hecke-Maass forms for SL(2,Z). Any uj(z) has a Fourier expansion of the form Also we let be the normalized L-function, which satisfies a functional equation relating s to 1 - s. Let g(z) be a holomorphic Hecke eigenform for SL(2, Z) of even integral weight k, with normalized Fourier coefficientsλg(n). The Rankin-Selberg L-function defined by is entire and also satisfies a functional equation relating s to 1 - s. Considering the weight sum of L(1/2,g x uj) and L(1/2,uj) over j, we will prove the following result.Theorem 1.1. Let g(z) be a fixed holomorphic Hecke eigenform of weight k, and {uj} an orthonormal basis of even Hecke-Maass forms of type 1/2+itj for SL(2, Z). Then for anyε>0 and T1/3+ε≤M≤T1/2, we have whereΣ’ means that the summation goes through the orthonormal basis of even Hecke-Maass forms, andThe main term in (0.11) can be compared with that in Theorem 1.1 of [29]. We remark that log T comes from the double pole when moving the integration line. To prove the theorem, we will use a strategy used by Li [29]. The proof starts with the approximate functional equations, which are yielded by the functional equations. Then we go via the Kuznetsov trace formula to the Kloosterman sum. Instead of using Weil’s bound, we open the Kloosterman sum and make crucial use of the Poisson summation formula and the Voronoi summation formula, so that we can use the stationary phase method twice.Using Theorem 1.1 together with the results of [10] and [7], we will prove the following assertion. Theorem 1.2. Under the same assumption as in Theorem 1.1, there are infinitely many u’js such that The above result states that L(s, g×uj) and L(s, uj) can be simultaneous nonvanishing at the central point s=1/2 infinite times.Besides, we will also prove the following bound. Theorem 1.3. We haveNote that the convexity bound of L(1/2, g×uj)L(1/2, uj) is《(1+|tj|)3/2+ε, so the above result is a subconvexity bound of the product of the two L-functions. For L(1/2,uj), the current record subconvexity bound is which was first proved conditionally by Iwaniec in [14], and an unconditional proof was given by Ivic [13] and subsequently by Jutila [21]. The best-known subconvexity bound for L(1/2, g×uj) was established by Lau, Liu and Ye in [28]: Denote by C(uj) and C(g×uj) the analytic conductors of L(s,uj) and L(s,g x uj), respectively. Then, according to Iwaniec-Sarnak [19], C(uj)(?)(1＋|tj|)2,C(g×Uj)(?)(1+|tj|)4. Thus, the quality of the bounds in (0.14) and (0.15) is the same strength as Weyl’s bound for the Riemann zeta-function in the sense that they reach C(uj)1/6+εand C(g×uj)1/6+ε, respectively. However, the goal of the present paper is not to improve the above two individual bounds. Instead, Using Theorem 1.1 together with the results in [28] and [21], we establish the following result on simultaneous nonvanishing of L(1/2, g x uj) and L(1/2,uj). Theorem 1.4. Under the same assumption as in Theorem 1.1, we have Note that the Weyl law states that where c is a constant. So the best possible result for (0.16) corresponds toε=0. |
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