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

## Abstract

We give a simple proof of a standard zero-free region in the *t*-aspect for the Rankin–Selberg *L*-function \(L(s,\pi \times \widetilde{\pi })\) for any unitary cuspidal automorphic representation \(\pi \) of \(\mathrm {GL}_n(\mathbb {A}_F)\) that is tempered at every nonarchimedean place outside a set of Dirichlet density zero.

## Keywords

Cuspidal automorphic representation Rankin–Selberg Zero-free region## Mathematics Subject Classification

11M26 (primary) 11F66 11N36 (secondary)## 1 Introduction

*F*be a number field, let

*n*be a positive integer, and let \(\pi \) be a unitary cuspidal automorphic representation of \(\mathrm {GL}_n(\mathbb {A}_F)\) with

*L*-function \(L(s,\pi )\), with \(\pi \) 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 \(\zeta (s)\) of the form

*L*-function having a zero of higher order than the order of the pole at \(s = 1\), then using an effective version of Landau’s lemma [12, Lemma 5.9].

Now let \(\pi '\) be a unitary cuspidal automorphic representation of \(\mathrm {GL}_{n'}(\mathbb {A}_F)\), and consider the Rankin–Selberg *L*-function \(L(s,\pi \times \pi ')\). Via the Langlands–Shahidi method, this extends meromorphically to the entire complex plane with at most a simple pole at \(s = 1\), with this pole occurring precisely when \(\pi ' \cong \widetilde{\pi }\). Moreover, this method shows that \(L(s,\pi \times \pi ')\) is nonvanishing in the closed right half-plane \(\mathfrak {R}(s) \ge 1\) [25, Theorem].

### Remark 1.2

One can also obtain the nonvanishing of \(L(s,\pi \times \pi ')\) on the line \(\mathfrak {R}(s) = 1\) via the Rankin–Selberg method. For \(n = n'\) and \(\pi ' \ncong \widetilde{\pi }\), this is shown in [21, Theorem 6.1]; the method of proof nonetheless is equally valid for \(n \ne n'\) or \(\pi ' \cong \widetilde{\pi }\), noting in the latter case that \(L(s,\pi \times \widetilde{\pi })\) has a simple pole at \(s = 1\) (see also [24, Equation (1.5)]). Note, however, that the product of *L*-functions considered in [21, Remark, p. 198] may *not* be used to show the desired nonvanishing of \(L(1 + it,\pi \times \pi ')\), but merely the nonvanishing of \(L(1,\pi \times \pi ')\).

Proving zero-free regions for \(L(s,\pi \times \widetilde{\pi })\) akin to (1.1), on the other hand, seems to be much more challenging. The method of de la Valleé–Poussin relies on the fact that the Rankin–Selberg convolutions \(L(s,\pi \times \pi )\) and \(L(s,\pi \times \widetilde{\pi })\) exist and extend meromorphically to \(\mathbb {C}\) with at most a simple pole at \(s = 1\). For \(L(s,\pi \times \pi ')\), the associated Rankin–Selberg convolutions have yet to be proved to have these properties, so as yet this method is inapplicable.

### Remark 1.3

Note that in [12, Exercise 4, p. 108], it is claimed that one can use this method to prove a zero-free region similar to (1.1) when \(\pi ' \ncong \pi \) and \(\pi ' \ncong \widetilde{\pi }\); however, the hint to this exercise is invalid, as the Dirichlet coefficients of the logarithmic derivative of the auxiliary *L*-function suggested in this hint are real but not necessarily nonpositive. (In particular, as stated, [12, Exercise 4, p. 108] would imply the nonexistence of Landau–Siegel zeroes upon taking *f* to be a quadratic Dirichlet character and *g* to be the trivial character.)

### Remark 1.4

When at least one of \(\pi \) and \(\pi '\) is self-dual, then this method *can* be used to prove a zero-free region akin to (1.1). When both \(\pi \) and \(\pi '\) are self-dual, this is proved by Moreno [21, Theorem 3.3] (see also [24, Equation (1.6)]). When only one of \(\pi \) and \(\pi '\) is self-dual, such a zero-free region has been stated by various authors (in particular, see [5, p. 619], [6, p. 92], and [7, p. 1]); to the best of our knowledge, however, no proof of this claim has appeared in the literature. In the appendix to this article written by Farrell Brumley, a complete proof of this result is given.

*t*| is sufficiently large; their method applies not only to automorphic representations of \(\mathrm {GL}_n(\mathbb {A}_F)\) but to more general reductive groups.

*t*-aspect. For \(\pi ' \ncong \widetilde{\pi }\), this is of the form

*s*in this zero-free region, while for \(\pi ' \cong \widetilde{\pi }\), this is of the form

*s*in this zero-free region.

*t*-aspect of a particular case of Brumley’s result, namely the case \(\pi ' \cong \widetilde{\pi }\) subject to the restriction that \(F = \mathbb {Q}\) and that \(\pi \) is unramified and tempered at every nonarchimedean place outside a set of Dirichlet density zero. With these assumptions, they prove the lower bound

In this article, we give a simple proof of the following.

### Theorem 1.9

*n*and

*F*) such that \(L(s,\pi \times \widetilde{\pi })\) has no zeroes in the region

*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 \(F = \mathbb {Q}\) and that \(\pi \) is unramified at every place. Nonetheless, we still require that \(\pi \) 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 \(L(s,\pi \times \widetilde{\pi })\) is the Riemann zeta function, so that \(F = \mathbb {Q}\) and \(\pi \) is trivial, and this method can also be adapted to prove a standard zero-free region in the *q*-aspect for \(L(s,\chi )\), where \(\chi \) 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 \(L(s,\pi \times \pi ')\) with \(\pi ' \ncong \widetilde{\pi }\), even in the particular case \(\pi ' = 1\) and \(n \ge 2\), so that \(L(s,\pi \times \pi ')\) is the standard *L*-function \(L(s,\pi )\). 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

Though this is a weaker result than Theorem 1.9, the method of proof is of particular interest; it is essentially a generalisation from \(\mathrm {GL}_1(\mathbb {A}_{\mathbb {Q}})\) to \(\mathrm {GL}_n(\mathbb {A}_F)\) 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 \(L(1 + it,\pi \times \widetilde{\pi })\) 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 \(L(1 + it, \pi \times \widetilde{\pi })\) and \(L'(1 + it, \pi \times \widetilde{\pi })\), 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 \(L(1 + it, \pi \times \widetilde{\pi })\) and \(L'(1 + it, \pi \times \widetilde{\pi })\); lower bounds for this smoothed average stem once again from sieve theory.

## 2 Sieve theory

*L*-function \(L(s,\pi )\) of \(\pi \) can be written as the Dirichlet series

*L*-function \(L(s,\pi \times \widetilde{\pi })\) is meromorphic on \(\mathbb {C}\) with only a simple pole at \(s = 1\). We denote by \(\Lambda _{\pi \times \widetilde{\pi }}(\mathfrak {a})\) the coefficients of the Dirichlet series for \(-\frac{L'}{L}(s,\pi \times \widetilde{\pi })\), so that

We denote by \(S_{\pi }\) the set of places of *F* at which \(\pi \) is either ramified or nontempered.

### Lemma 2.1

### Proof

*m*, and so via the bound \(|\sin x| \le |x|\), we have that

*C*sufficiently small in terms of

*n*and

*F*, (2.3) and (2.5) imply that

### Remark 2.6

The only point at which we make use of the assumption that \(\pi \) 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 \(\pi \), much of the argument still works if we keep track of this dependence in terms of the analytic conductor of \(\pi \). 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 \(L'(\sigma + it, \pi \times \widetilde{\pi })\) 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 \(\pi \), but the payoff would not be great as the weaker zero-free region (1.5) would only give a weak error term.

## 3 Proof of Theorem 1.9

*L*-function of \(\Pi \) and \(\widetilde{\Pi }\) factorises as

### Lemma 3.2

### Proof

### Lemma 3.3

### Proof

Now let us restrict to the case \(\pi ' = \widetilde{\pi }\).

### Corollary 3.4

### Proof

### Proof of Theorem 1.9

### Remark 3.5

## 4 Proof of Theorem 1.12

*s*near 1 and

*s*near \(1 \pm it\). Finally, we write

*s*near 1.

### Lemma 4.2

### Proof

This follows from the factorisation \(L(s,\pi \times \widetilde{\pi }) = \zeta _F(s) L(s, {{\mathrm{ad}}}\pi )\). \(\square \)

### Lemma 4.3

### Proof

This is proved by Goldfeld and Li in [7, Lemma 5.1] for \(F = \mathbb {Q}\) and \(\pi ' \cong \widetilde{\pi }\), but the proof in this more generalised setting (via the approximate functional equation) follows mutatis mutandis. \(\square \)

Together with the fact that \(L(1,{{\mathrm{ad}}}\pi ) \ne 0\), this shows the following.

### Corollary 4.4

### 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.

### Lemma 4.7

### Proof

*n*-powers and using the fact that \(\lambda _{\Pi \times \widetilde{\Pi }}(\mathfrak {a}^{2n}) \ge 1\), our approach is to restrict to prime ideals at which \(\pi \) is unramified and tempered and use sieve theory to show that \(\lambda _{\Pi \times \widetilde{\Pi }}(\mathfrak {p})\) is often not too small on dyadic intervals.

### Lemma 4.8

### Proof

This follows via the same method as the proof of Lemma 3.3. \(\square \)

We now restrict to the case \(\pi ' = \widetilde{\pi }\).

### Proof of Theorem 1.12

## Footnotes

- 1.
Supported by ANR grant 14-CE25.

- 2.
I would like thank Philippe Michel and Étienne Fouvry for suggesting that I write up a proof of Theorem A.1. I am grateful as well to Peter Humphries for allowing me to include this appendix to his paper, and for suggesting many improvements to the proofs and exposition.

## Notes

### Acknowledgements

The author would like to thank Peter Sarnak, Dorian Goldfeld, and Farrell Brumley for helpful discussions and comments, as well as Dimitris Koukoulopoulos for alerting the author to an error in an earlier version of this article.

## References

- 1.Balasubramanian, R., Ramachandra, K.: The place of an identity of Ramanujan in prime number theory. Proc. Indian Acad. Sci. Sect. A
**83**(4), 156–165 (1976). https://doi.org/10.1007/BF03051376 MathSciNetCrossRefzbMATHGoogle Scholar - 2.Banks, W.D.: Twisted symmetric square \(L\)-functions and the non-existence of siegel zeros on \({{\rm GL}}(3)\). Duke Math. J.
**87**(2), 343–353 (1997). https://doi.org/10.1215/S0012-7094-97-08713-5 MathSciNetCrossRefzbMATHGoogle Scholar - 3.Brumley, F.: Effective multiplicity one on \({{\rm GL}}(n)\) and zero-free regions of Rankin–Selberg \(L\)-functions. Am. J. Math.
**128**(6), 1455–1474 (2006). https://doi.org/10.1353/ajm.2006.0042 MathSciNetCrossRefzbMATHGoogle Scholar - 4.Bushnell, C.J., Henniart, G.: An upper bound on conductors for pairs. J. Number Theory
**65**(2), 183–196 (1997). https://doi.org/10.1006/jnth.1997.2142 MathSciNetCrossRefzbMATHGoogle Scholar - 5.Gelbart, S.S., Lapid, E.M.: Lower bounds for \(L\)-functions at the edge of the critical strip. Am. J. Math.
**128**(3), 619–638 (2006). https://doi.org/10.1353/ajm.2006.0024 MathSciNetCrossRefzbMATHGoogle Scholar - 6.Gelbart, S.S., Lapid, E.M., Sarnak, P.: A new method for lower bounds of \(L\)-functions. Comptes Rendus de l’Académie des Sciences. Série 1, Mathématique
**339**(2), 91–94 (2004). https://doi.org/10.1016/j.crma.2004.04.024 MathSciNetCrossRefzbMATHGoogle Scholar - 7.Goldfeld, D., Li, X.: A standard zero free region for Rankin Selberg \(L\)-functions. Int. Math. Res. Not. (2017). https://doi.org/10.1093/imrn/rnx087 Google Scholar
- 8.Goldfeld, D., Hoffstein, J., Lieman, D.: “An effective zero-free region”, appendix to “coefficients of maass forms and the siegel zero” by Jeffrey Hoffstein and Paul Lockhart. Ann. Math.
**140**(1), 177–181 (1994). https://doi.org/10.2307/2118544 CrossRefGoogle Scholar - 9.Grenié, L., Molteni, G., Perelli, A.: Primes and prime ideals in short intervals. Mathematika
**63**(2), 364–371 (2017). https://doi.org/10.1112/S0025579316000310 MathSciNetCrossRefzbMATHGoogle Scholar - 10.Hoffstein, J., Ramakrishnan, D.: Siegel zeros and cusp forms. Int. Math. Res. Not.
**1995**(6), 279–308 (1995). https://doi.org/10.1155/S1073792895000225 MathSciNetCrossRefzbMATHGoogle Scholar - 11.Humphries, P.: Effective lower bounds for \(L(1,\chi )\) via Eisenstein series. Pac. J. Math.
**288**(2), 355–375 (2017). https://doi.org/10.2140/pjm.2017.288.355 MathSciNetCrossRefzbMATHGoogle Scholar - 12.Iwaniec, H., Kowalski, E.: Analytic Number Theory, American Mathematical Society Colloquium Publications 53. American Mathematical Society, Providence (2004). https://doi.org/10.1090/coll/053 Google Scholar
- 13.Iwaniec, H., Sarnak, P.: Perspectives on the Analytic Theory of \(L\)-Functions. In: Alon, N., Bourgain, J., Connes, A., Gromov, M., Milman, V. (eds.) Vision in Mathematics. GAFA 2000 Special Volume, Part II, pp. 705–741. Birkhäuser, Basel (2000). https://doi.org/10.1007/978-3-0346-0425-3_6 Google Scholar
- 14.Jacquet, H., Shalika, J.A.: On Euler products and the classification of automorphic representations I. Am. J. Math.
**103**(3), 499–558 (1981). https://doi.org/10.2307/2374103 MathSciNetCrossRefzbMATHGoogle Scholar - 15.Kim, H.H.: Functoriality for the exterior square of \({{\rm GL}}_4\) and the symmetric fourth of \({{\rm GL}}_2\)”, with appendix 1 by Dinakar Ramakrishnan and appendix 2 by Henry H. Kim and Peter Sarnak. J. Am. Math. Soc.
**16**(1), 139–183 (2003). https://doi.org/10.1090/S0894-0347-02-00410-1 CrossRefzbMATHGoogle Scholar - 16.Kim, H.H., Shahidi, F.: Cuspidality of symmetric powers with applications. Duke Math. J.
**112**(1), 177–197 (2002). https://doi.org/10.1215/S0012-9074-02-11215-0 MathSciNetCrossRefzbMATHGoogle Scholar - 17.Knapp, A.W.: Local langlands correspondence: the archimedean case in motives. In: Jannsen, U., Kleiman, S.L., Serre, J.-P. (eds.) Proceedings of Symposia in Pure Mathematics vol. 55, no(2), pp. 393–410. American Mathematical Society, Providence (1994). https://doi.org/10.1090/pspum/055.2 Google Scholar
- 18.Lapid, E.: “On the Harish–Chandra Schwartz space of \(G(F) \backslash G({\mathbb{A}})\)”, with an appendix by Farrell Brumley. In: Prasad, D., Rajan, C.S., Sankaranarayanan, A., Sengupta, J. (eds.) Automorphic Representations and \(L\)-Functions. Proceedings of the International Colloquium, Mumbai 2012, pp. 335–377. Hindustan Book Agency, New Delhi (2013)Google Scholar
- 19.Li, X.: Upper bounds for \(L\)-functions at the edge of the critical strip. Int. Math. Res. Not.
**2010**(4), 727–755 (2010). https://doi.org/10.1093/imrn/rnp148 MathSciNetzbMATHGoogle Scholar - 20.Moreno, C.J.: Explicit formulas in the theory of automorphic forms. In: Melvyn, N.B. (ed.) Number Theory Day, Lecture Notes in Mathematics, vol. 626, pp. 73–216. Springer, Berlin (1977)Google Scholar
- 21.Moreno, C.J.: Analytic proof of the strong multiplicity one theorem. Am. J. Math.
**107**(1), 163–206 (1985). https://doi.org/10.2307/2374461 MathSciNetCrossRefzbMATHGoogle Scholar - 22.Ramakrishnan, D., Wang, S.: On the exceptional zeros of Rankin–Selberg \(L\)-functions. Compositio Mathematica
**135**(2), 211–244 (2003). https://doi.org/10.1023/A:1021761421232 MathSciNetCrossRefzbMATHGoogle Scholar - 23.Rudnick, Z., Sarnak, P.: Zeros of principal \(L\)-functions and random matrix theory. Duke J. Math.
**81**(2), 269–322 (1996). https://doi.org/10.1215/S0012-7094-96-08115-6 MathSciNetCrossRefzbMATHGoogle Scholar - 24.Sarnak, P.: Nonvanishing of \(L\)-functions on \(\Re (s) = 1\). In: Hida, H., Ramakrishnan, D., Shahidi, F. (eds.) Contributions to Automorphic Forms, Geometry and Number Theory, pp. 719–732. The John Hopkins University Press, Baltimore (2004)Google Scholar
- 25.Shahidi, F.: On nonvanishing of \(L\)-functions. Bull. Am. Math. Soc.
**2**(3), 462–464 (1980). https://doi.org/10.1090/S0273-0979-1980-14769-2 MathSciNetCrossRefzbMATHGoogle Scholar - 26.Titchmarsh, E.C.: The Theory of the Riemann Zeta-function, 2nd edn. Clarendon Press, Oxford (1986)zbMATHGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.