Standard zero-free regions for Rankin–Selberg L-functions via sieve theory

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Introduction
Let F be a number field, let n be a positive integer, and let π be a unitary cuspidal automorphic representation of GL n (A F ) with L-function L(s, π), with π normalised such that its central character is trivial on the diagonally embedded copy of the positive reals. The proof of the prime number theorem due to de la Valleé-Poussin gives a zero-free region for the Riemann zeta function ζ(s) of the form σ > 1 − c log(|t| + 3) for s = σ + it, and this generalises to a zero-free region for L(s, π) of the form for some positive constants c π,π , N π,π dependent on π and π , provided that |t| is sufficiently large; their method applies not only to automorphic representations of GL n (A F ) but to more general reductive groups.
Recently, Goldfeld and Li [7] have given a strengthening in the t-aspect of a particular case of Brumley's result, namely the case π ∼ = π subject to the restriction that F = Q and that π is unramified and tempered at every nonarchimedean place outside a set of Dirichlet density zero. With these assumptions, they prove the lower bound L(1 + it, π × π) π 1 (log(|t| + 3)) 3 (1.7) for |t| ≥ 1, which gives a zero-free region of the form σ ≥ 1 − c π (log(|t| + 3)) 5 (1. 8) for some positive constant c π dependent on π provided that |t| ≥ 1. Their proof, like that of Gelbart and Lapid [5], makes use of Sarnak's effectivization of the Langlands-Shahidi method; the chief difference is that, like Sarnak but unlike Gelbart and Lapid, they are able to use sieve theory to obtain a much stronger zero-free region. On the downside, the proof is extremely long and technical, and, being written in the classical language instead of the adèlic language, any generalisation of their method to arbitrary number fields and allowing ramification of π would be a challenging endeavour. (Indeed, the Langlands-Shahidi method, in practice, is rather inexplicit at ramified places, though see [11] for explicit calculations for the case n = 1 and F = Q, so that π corresponds to a primitive Dirichlet character.) In this article, we give a simple proof of the following. Theorem 1.9 Let π be a unitary cuspidal automorphic representation of GL n (A F ) that is tempered at every nonarchimedean place outside a set of Dirichlet density zero. Then there exists an absolute constant c π dependent on π (and hence also on n and F) such that L(s, π × π) has no zeroes in the region with |t| ≥ 1. Furthermore, we have the bound L(s, π × π) π 1 log(|t| + 3) (1.11) for s in this region.
In particular, we improve the zero-free region (1.8) and lower bound (1.7) of Goldfeld and Li to (1.10) and (1.11) respectively while removing Goldfeld and Li's restriction that F = Q and that π is unramified at every place. Nonetheless, we still require that π be tempered at every nonarchimedean place outside a set of Dirichlet density zero; moreover, this zero-free region is only in the t-aspect, unlike Brumley's zero-free region in the analytic conductor aspect.
The proof of Theorem 1.9 shares some similarities with the method of de la Valleé-Poussin. Once again, one creates an auxiliary L-function, though this has a zero of equal order to the order of the pole at s = 1. While Landau's lemma cannot be used in this setting to obtain a standard zero-free region, one can instead use sieve theory. This approach is discussed in [26,Section 3.8] when L(s, π × π) is the Riemann zeta function, so that F = Q and π is trivial, and this method can also be adapted to prove a standard zero-free region in the q-aspect for L(s, χ), where χ is a primitive Dirichlet character; cf. [1,11].
This usage of sieve theory, however, seems to have limitations; it is unclear how to prove zero-free regions via sieve theory for Rankin-Selberg L-functions L(s, π ×π ) with π π, even in the particular case π = 1 and n ≥ 2, so that L(s, π × π ) is the standard L-function L(s, π). We expand upon this point in Remark 3.5.
By slightly different means, we sketch how to prove a weaker version of Theorem 1.9.

Theorem 1.12
Let π be a unitary cuspidal automorphic representation of GL n (A F ) that is tempered at every nonarchimedean place outside a set of Dirichlet density zero. Then for |t| ≥ 1, we have the bound 13) and so there exists an absolute constant c π dependent on π such that L(s, π × π) has no zeroes in the region σ ≥ 1 − c π (log(|t| + 3)) 5 . (1.14) Though this is a weaker result than Theorem 1.9, the method of proof is of particular interest; it is essentially a generalisation from GL 1 (A Q ) to GL n (A F ) of the method of Balasubramanian and Ramachandra [1]. It turns out that Brumley's method [3] in proving (1.6) is a natural generalisation of [1] except that sieve theory is not used and so the resulting lower bounds for L(1 + it, π × π) are not nearly as strong.
Theorem 1.12 gives the same bounds as obtained by Goldfeld and Li, and this is no accident. Goldfeld and Li create an integral of an Eisenstein series and obtain upper bounds for this integral via the Maaß-Selberg relation together with upper bounds for L(1+it, π × π) and L (1 + it, π × π), while they use the Fourier expansion of the Eisenstein series together with sieve theory to find lower bounds for this integral. In the proof of Theorem 1.12, we follow Brumley's method of studying a smoothed average of the Dirichlet coefficients of an auxiliary L-function. Upper bounds for this smoothed average are then obtained via Perron's inversion formula and Cauchy's residue theorem, in place of Goldfeld and Li's usage of the Maaß-Selberg relation, together with upper bounds for L(1+it, π × π) and L (1+it, π × π); lower bounds for this smoothed average stem once again from sieve theory.

Sieve theory
The L-function L(s, π) of π can be written as the Dirichlet series for (s) sufficiently large, where N (a) = N F/Q (a) := #O F /a, and extends to a meromorphic function on C with at most a simple pole at s = 1 if n = 1 and π is trivial, so that L(s, π) = ζ F (s). Similarly, the Rankin-Selberg L-function L(s, π × π) is meromorphic on C with only a simple pole at s = 1. We denote by π × π (a) the coefficients of the Dirichlet series for − L L (s, π × π), so that These coefficients are nonnegative; see [12,Remark,p. 138]. Moreover, the residue of this at s = 1 is 1, and we have that whenever π is unramified at p. We denote by S π the set of places of F at which π is either ramified or nontempered.
Proof We use Ikehara's Tauberian theorem and the fact that S π has Dirichlet density zero to see that The assumption that π is tempered at every nonarchimedean place outside a set of Dirichlet density zero implies that |λ π (p)| ≤ n whenever p / ∈ S π , so that for any C > 0, the left-hand side of (2.2) is and as we ascertain that Next, for C ∈ (0, 2), we note that for any integer m, and so via the bound | sin x| ≤ |x|, we have that From [9, Proposition 2], we have that the proof of this reduces to the case F = Q, in which case this is a well-known result that can be proven via the Selberg sieve (with the appearance of an additional error term) or the large sieve. So assuming that 1 and Y > 4|t| 2 , the inner term on the right-hand side of (2.4) is bounded by By choosing C sufficiently small in terms of n and F, (2.3) and (2.5) imply that from which the result follows.

Remark 2.6
The only point at which we make use of the assumption that π is tempered at every nonarchimedean place outside a set of Dirichlet density zero is in proving (2.3). It would be of interest whether an estimate akin to (2.3) could be proved unconditionally.

Remark 2.7
While the implicit constants in Theorems 1.9 and 1.12 depend on π, much of the argument still works if we keep track of this dependence in terms of the analytic conductor of π. The main issue seems to be the lower bound stemming from Lemma 2.1; in particular, the use of Ikehara's Tauberian theorem to prove (2.2). We could instead use (1.6) together with an upper bound for L (σ + it, π × π) in the region (1.5) derived via the methods of Li [19] to prove (2.2) with an error term that is effective in terms of the analytic conductor of π, but the payoff would not be great as the weaker zero-free region (1.5) would only give a weak error term.

Proof of Theorem 1.9
Let π and π be unitary cuspidal automorphic representations of GL n (A F ) and GL n (A F ) respectively. Let ρ = β + iγ be a nontrivial zero of L(s, π × π ) with 1/2 ≤ β < 1 and γ = 0. We define This is an isobaric (noncuspidal) automorphic representation of GL n+n (A F ). The Rankin-Selberg L-function of and factorises as This is a meromorphic function on C with a double pole at s = 1, simple poles at s = 1±iγ if π ∼ = π, and holomorphic elsewhere. We let × (a) denote the coefficients of the Dirichlet Again, these coefficients are nonnegative.

Lemma 3.3
Suppose that π and π are unramified at p. Then Proof Indeed, (3.1) implies that whenever π and π are unramified at p. Now let us restrict to the case π = π. Corollary 3.4 Suppose that π is tempered at every nonarchimedean place outside a set of Dirichlet density zero. Then for σ > 1, by dividing into dyadic intervals and applying Lemma 2.1.
Finally, we note that which is equivalent to (1.11).

Remark 3.5
To prove Theorem 1.9 for L(s, π × π ) with π π, we would need to replace Lemma 2.1 with a result of the form but it is unclear how one might generalise the proof of Lemma 2.1 to obtain such a result.

Proof of Theorem 1.12
For t ∈ R\{0}, define the isobaric automorphic representation of GL n+n (A F ) by Then This is a meromorphic function on C with a double pole at s = 1, simple poles at s = 1 ± it if π ∼ = π, and holomorphic elsewhere. We let λ × (a) denote the coefficients of the Dirichlet series for L(s, × ), so that Again, the coefficients λ × (a) are nonnegative. Write for s near 1 and for s near 1 ± it. Finally, we write for s near 1.
Together with the fact that L(1, ad π) = 0, this shows the following.
Remark 4.6 While (4.5) is stated in [12, (5.11)], a complete proof does not seem to have appeared in the literature. In the appendix to this article, a proof of (a more general version of) this statement is given.

L(s, × ) ψ(s)Y s ds
for any σ < 1. The convexity bound of Li [19] and the rapid decay of ψ in vertical strips imply that from which the result follows.
In [3,18], Brumley notes that and that λ × (a 2n ) ≥ 1. This is paired with a modified version of Lemma 4.7 in order to prove effective lower bounds for |L(1 + it, π × π)| in terms of the analytic conductor q(π × π, 1+it). Instead of restricting the sum over integral ideals to those that are 2n-powers and using the fact that λ × (a 2n ) ≥ 1, our approach is to restrict to prime ideals at which π is unramified and tempered and use sieve theory to show that λ × (p) is often not too small on dyadic intervals.

Lemma 4.8
Suppose that π and π are unramified at p. Then Proof This follows via the same method as the proof of Lemma 3.3.
We now restrict to the case π = π .
Proof of Theorem 1.12 From Lemma 4.7 and Corollary 4.4, we have that for Y ≥ q( × ), there exists δ > 0 such that On the other hand, Lemmata 4.8 and 2.1 imply that for Y π (|t| + 3) 2 , This gives the lower bound (1.13). Then as in [7, Proof of Theorem 1.2], (1.14) follows via the mean value theorem.
The aim of this appendix is to provide a proof of the claim, stated in Gelbart-Lapid-Sarnak [6, p. 92] and Gelbart-Lapid [5, p. 619], that Rankin-Selberg L-functions are known to satisfy a standard zero-free region whenever at least one of the forms is self-dual. This is Theorem A.1 below. The method is through the classical argument of de la Vallée-Poussin. We take advantage of the occasion to clarify parts of the literature, and to justify, in Lemma A.2, another oft claimed inequality on the archimedean conductor. 2 Theorem A.1 Let n, n ≥ 1. Let F be a number field. Let π and π be unitary cuspidal automorphic representations of GL n (A F ) and GL n (A F ), respectively. We normalize π and π so that their central characters are trivial on the diagonally embedded copy of the positive reals. Assume that π is self-dual. There is an effective absolute constant c > 0 such that L(s, π × π ) is non-vanishing for all s = σ + it ∈ C verifying with the possible exception of one real zero whenever π is also self-dual.
Remark A. 1 In [22] it is shown that when n = n = 2, L(s, π ×π ) satisfies the conclusions of Theorem A.1, except possibly when it is divisible by the L-function of a quadratic character. A similar result holds for n = n = 3 when π and π are symmetric square lifts from self-dual forms on GL 2 .
Remark A.2 Theorem A.1 implies a standard zero-free region for the L-function of a nonself dual cusp form π on GL n , by taking π to be the trivial character on GL 1 . The fact that L(s, π) admits no exceptional zeros whenever π is not self-dual is originally due to Moreno [20, Theorem 5.1] when n = 2 and Hoffstein-Ramakrishnan [10, Corollary 3.2] for n ≥ 3. (For complex characters it is classical.)

Remark A.3
If π is self-dual on GL n , then Theorem A.1 allows for the possibility of a single exceptional zero, necessarily real, of L(s, π). There are cases when this exceptional zero can be provably eliminated. To the best of our knowledge, these cases are, at the time of this writing, limited to the following situations: (1) when π is a cusp form on GL 2 , due to Hoffstein-Ramakrishnan [10, Theorem C]; (2) when π is a cusp form on GL 3 . This is due to Banks [2, Theorem 1], who verifies Hypothesis 6.1 in [10]. (3) when π is a cusp form on GL 5 which arises as the symmetric fourth power lift of a cusp form on GL 2 which is not of solvable polynomial type. This is due to Ramakrishnan-Wang [22]; see the comments after Corollary C in loc cit.; (4) for the L-functions L(s, π, sym 6 ) and L(s, π, sym 8 ), when π is a self-dual cusp from on GL 2 . This is Theorem D in [22].
All of these results build on the groundbreaking work of [8]. Moreover, cases (3) and (4) make full use of the advances in functoriality by Kim and Shahidi [15,16].
Remark A. 4 We emphasize the importance of the cuspidality condition in (1) and (2) in the above remark, which rules out the divisibility of L(s, π) by the L-function of a quadratic character. For example, if π is a dihedral form on GL 2 over F, induced by a Hecke character χ of a quadratic field extension E, then L(s, π) = L(s, χ). Now if π is cuspidal, χ does not factor through the norm, which (as was remarked in [22]) rules out χ real. One can then obtain a standard zero-free region for π by appealing to the classical GL 1 case for complex (Hecke) characters over E. The original argument given in [10,Theorem B and Remark 4.3] for dihedral forms on GL 2 is, on the surface, more complicated, but this is simply due to to the more general framework in which it is set.
Similarly, the cuspidality condition for GL 3 rules out the possibility that π arises as the symmetric square lift of a dihedral form on GL 2 .
All of the above remarks pertain to results in the full conductor aspect only; for the t-aspect, we refer to the body of the paper.
Let m ≥ 1 be the order of the pole of D(s) at s = 1. Then [12,Theorem 5.9] (which is based on [8,Lemma]) states that there is a constant κ > 0 such that D(s) has no more than m real zeros in the interval Let us calculate the integer m. The factor L 1 (s) has a pole of order 3 at s = 1. Moreover, if t = 0 the factor L 2 (s) is holomorphic at s = 1. When t = 0, the regularity of L 2 (s) at s = 1 depends on whether or not π is self-dual: (1) if π is not self-dual and t = 0, the function L 2 (s) is holomorphic at s = 1, since necessarily π = π and π = π ; (2) if π is self-dual and t = 0, the function L 2 (s) has a pole of order 2 or 4, according to whether π = π or π = π .
We deduce that m = 3 when either π is not self-dual or t = 0. When π is self-dual and t = 0, we have m = 5 or 7, according to whether π = π or π = π . Now let σ ∈ (0, 1) and suppose that L(s, π × π ) vanishes to order r at s = σ + it. By the functional equation and the self-duality of π , this is equivalent to L(σ − it, π × π ) vanishing to order r at s = σ − it. From this it follows that L 2 (s), and hence D(s), has a zero at s = σ of order 4r . Moreover, this is the case regardless of the value of t or whether π is self-dual. If σ lies in the range (A.5), then since 4r ≤ m, we deduce from the previous paragraph that r = 0 whenever π is not self-dual or t = 0, and r ≤ 1 whenever π is self-dual and t = 0.
The bounds of [4,Theorem 1], applied to the finite conductor of each factor above, yield For the archimedean conductor, Lemma A.2 below implies that for an absolute constant C 1 > 0. This yields for an absolute constant C 2 > 0, which finishes the proof.
The following result -the analog at archimedean places of the Bushnell-Henniart bounds (A.6) on the Rankin-Selberg conductor -has been claimed without proof in many sources, including [12, (5.11)] and our own [3] (to name just two). Nevertheless, there does not seem an available proof in the literature.
Lemma A.2 Let F v be R or C. Let n, n ≥ 1 be integers. Let π v and π v be irreducible unitary generic representations of GL n (F v ) and GL n (F v ), respectively. There is an absolute constant C > 0 such that Remark A. 8 The constant C in Lemma A.2 can be explicitly computed, and the proof gives an exact value. But since the archimedean conductor should not be considered an "exact quantity" (and conventions for the definition vary according to the source), it makes little sense to include the precise value of C in the estimate. Remark A. 9 In the course of the proof, we shall recall the definition of the archimedean Rankin-Selberg and standard analytic conductor as given by Iwaniec-Sarnak in [13]. Their ad hoc recipe boils down to taking the product over all Gamma shifts arising in the local L-factors. It will be apparent that the definition of q v (it; π × π ) can be made in the admissible (rather than unitary generic) dual. One may drop the hypothesis of genericity (but not unitarity) in the statement of Lemma A.2 at the price of allowing the constant C to depend linearly on n and n .
for some constants B, C > 0, depending on n and n . Indeed, the archimedean factor of the "thickened level" M(π) introduced in [loc. cit., Definition 1.4], is defined using the sum, rather than the product, of all Gamma shifts. (Note that in [8] the max of the Gamma shifts is taken.) Thus M v (π), for v | ∞, behaves quantitatively much differently than the archimedean factor of the analytic conductor of Iwaniec-Sarnak [13]. Since its appearance, the latter has become the preferred measure of complexity in the study of L-functions. It should be emphasized that since log q v (it; π × π ) log M v (it; π × π ) and log q v (it; π) log M v (it; π), the bounds (A.7) with unspecified exponents are consequences of the work of Hoffstein-Ramakrishnan. Thus the proof of Lemma A.1 can be made to be independent of Lemma A.2, at the price of an inexplicit dependence in the implied constant on n and n . In any case, the proof of Lemma A.2 is closely modelled on that of [10, Lemma b], with the appropriate modifications for dealing with analytic conductor.
Proof Using Langlands' classification of the admissible dual of GL n (F v ) (see, for example, [17]), π v and π v correspond to ⊕ϕ i and ⊕ϕ j , for irreducible representations ϕ i and ϕ j of the Weil group W F v of F v . By definition, we have which gives rise to similar factorizations of the associated conductors. Let d i , d j denote the dimensions of ϕ i and ϕ j , respectively, so that n = d i and n = d j . Dropping the indices i and j, we must therefore prove that for irreducible representations ϕ and ϕ of W F v , of respective dimensions d and d , we have for an absolute constant C > 0. When F v = C, one has W C = C × , so that all irreducible representations are onedimensional. We may write any such character as χ k,ν (z) = (z/|z|) k |z| 2ν , for k ∈ Z and ν ∈ C. Letting μ = ν + |k|/2, the associated L-factor (see [17, (4.6)]) is C (s + μ). The recipe of Iwaniec-Sarnak [13, (21) and (31)] gives Now if ϕ = χ k,ν and ϕ = χ k ,ν , then ϕ ⊗ ϕ = χ k+k ,ν+ν . This implies that An application of the triangle inequality yields We claim that |k| 2 + |ν| | |k| 2 + ν| = |μ|. We may assume that k = 0. On one hand, On the other, since π and π are unitary generic, the Jacquet-Shalika bounds |Re(ν)| ≤ we establish (A.12) in the case F v = C. When F v = R, each irreducible representation ϕ of W R = C × ∪ jC × is of dimension 1 or 2. If ϕ is one-dimensional, then its restriction to C × is χ 0,ν for ν ∈ C (see [17, (3.2)]). We let k = 1 − ϕ( j) ∈ {0, 2}. Writing μ = ν + k/2, we have L(s, ϕ) = R (s + μ) and q v (it, ϕ) = 1 + |it + μ|.
In either case, let (k, ν) and (k , ν ) be the parameters associated with ϕ and ϕ , respectively. We now examine the tensor products parameters.
This follows (with C = 1) from applying the triangle inequality and (A.13) to each factor on the left-hand side.
This completes the proof of Lemma A.2.