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

Humphries, Peter; Brumley, Farrell

doi:10.1007/s00209-018-2136-8

Cited by 31 publications

(18 citation statements)

References 24 publications

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“…Given π ∈ F n and π ′ ∈ F n ′ , let L(s, π × π ′ ) be the associated Rankin-Selberg L-function, and let π ∈ F n and π ′ ∈ F n ′ be the contragredient representations. When π ′ ∈ { π, π ′ }, work of Brumley [29,Appendix] and the authors [30] shows that there exists an absolute and effectively computable constant c 1 > 0 such that L(s, π × π ′ ) has a "standard" zero-free region of the shape…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…If π ′ ∈ { π, π ′ }, then L(s, π × π ′ ) has the standard zero-free region (1.1) apart from a possible Landau-Siegel zero, which must be both real and simple. In all other cases, it follows from work of Brumley (see [8], [29,Section 1], and [41,Appendix]) that there exists an absolute and effectively computable constant c 2 > 0 such that L(s, π × π ′ ) = 0 in the much narrower region (2.1) Re(s)…”

Section: 2mentioning

confidence: 99%

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Zeros of Rankin-Selberg $L$-functions in families

Humphries,

Thorner

2021

Preprint

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Let F n be the set of all cuspidal automorphic representations π of GL n with unitary central character over a number field F . We prove the first unconditional zero density estimate for the set S = {L(s, π × π ′ ) : π ∈ F n } of Rankin-Selberg L-functions, where π ′ ∈ F n ′ is fixed. We use this density estimate to prove:(i) a strong average form of effective multiplicity one for GL n ;(ii) that given π ∈ F n defined over Q, the convolution π × π has a positive level of distribution in the sense of Bombieri-Vinogradov; (iii) that almost all L(s, π×π ′ ) ∈ S have a hybrid-aspect subconvexity bound on Re(s) = 1 2 ; (iv) a hybrid-aspect power-saving upper bound for the variance in the discrepancy of the measures |ϕ(x + iy)| 2 y −2 dxdy associated to GL 2 Hecke-Maaß newforms ϕ with trivial nebentypus, extending work of Luo and Sarnak for level 1 cusp forms; and (v) a nonsplit analogue of quantum ergodicity: almost all restrictions of Hilbert Hecke-Maaß newforms to the modular surface dissipate as their Laplace eigenvalues grow.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Zeros of Rankin-Selberg $L$-functions in families

Humphries,

Thorner

2021

Preprint

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“…and by the bounds for 1 ( ), 2 ( ) given in (7.0.12), P 2,2 ,Φ ( , 1, 1), = Note that the bound for the case that |Im( 1 )| is small involves the non-existence of Siegel zeros that is proved in [HR95] while the case when |Im( 1 )| is large was first proved in [Mor85]. See also [GL18], [HB19]. Applying these bounds, the Theorem 7.0.3 bound, together with Stirling's bound to estimate the integral in 1 , we find, as claimed, that…”

Section: Bounding the Contribution From The Continuous Spectrummentioning

confidence: 95%

An orthogonality relation for (with an appendix by Bingrong Huang)

Goldfeld

Stade

Woodbury

2021

Forum of Mathematics, Sigma

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Orthogonality is a fundamental theme in representation theory and Fourier analysis. An orthogonality relation for characters of finite abelian groups (now recognized as an orthogonality relation on $\mathrm {GL}(1)$ ) was used by Dirichlet to prove infinitely many primes in arithmetic progressions. Orthogonality relations for $\mathrm {GL}(2)$ and $\mathrm {GL}(3)$ have been worked on by many researchers with a broad range of applications to number theory. We present here, for the first time, very explicit orthogonality relations for the real group $\mathrm {GL}(4, \mathbb R)$ with a power savings error term. The proof requires novel techniques in the computation of the geometric side of the Kuznetsov trace formula.

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“…Such a uniform bound is known to be a consequence of the Rankin–Selberg theory. However, it is apparent that the first proof of this fact in the literature is given by Brumley [, Lemma A.2]. Also, we do require the bound of the Rankin–Selberg conductors

A_{π \otimes \overset{π}{true} \otimes χ}

established by Bushnell and Henniart .…”

Section: Applications To Automorphic L‐functionsmentioning

confidence: 99%

“…To end this section, we further remark that the prime number theorem and log‐free zero‐density estimates for automorphic L‐functions have been studied in . Moreover, it is possible to obtain the error terms for the above estimates by invoking the zero‐free regions of the Rankin–Selberg L‐functions (see, e.g., ).…”

Section: Applications To Automorphic L‐functionsmentioning

confidence: 99%

Bombieri–vinogradov Theorems for Modular Forms and Applications

Wong

2019

Mathematika

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In this article, we consider a prime number theorem for arithmetic progressions “weighted” by Fourier coefficients of modular forms, and we develop Siegel‐Walfisz type and Bombieri–Vinogradov type estimates for such a modular analogue. As an application, we have a Turán type estimate for modular forms asserting that for any δ>0 and non‐CM normalised Hecke eigenform f, scriptPffalse(a,qfalse)≤q2+δ,with a possible exceptional set of q of density 0 (depending at most on f and δ), where (a,q)=1, scriptPffalse(a,qfalse) denotes the least prime p, with λffalse(pfalse)≠0, congruent to a(modq), and λffalse(pfalse) is the pth Fourier coefficient of f. Moreover, we show the existence of a positive absolute constant C0, independent of f, such that there are infinitely many pairs (p1,p2) of distinct primes satisfying false|p1−p2false|≤C0andλffalse(p1false)λffalse(p2false)≠0,which presents a modular analogue of the recent work of Maynard and Zhang on bounded gaps between primes.

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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, π × π) for any unitary cuspidal automorphic representation π of GL n (A F ) that is tempered at every nonarchimedean place outside a set of Dirichlet density zero.

Cited by 31 publications

References 24 publications

Zeros of Rankin-Selberg $L$-functions in families

Zeros of Rankin-Selberg $L$-functions in families

An orthogonality relation for (with an appendix by Bingrong Huang)

Bombieri–vinogradov Theorems for Modular Forms and Applications

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