Surpassing the ratios conjecture in the 1-level density of Dirichlet<i>L</i>-functions

Fiorilli, Daniel; Miller, Steven J.

doi:10.2140/ant.2015.9.13

Cited by 25 publications

(24 citation statements)

References 63 publications

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“…Thus, if 2g 2K+1 ≤ N ≤ 2g−1 2K−1 for some K ≥ 1, or 2g ≤ N < 4g (corresponding to K = 0), then we have (9) and the previous subsection that we can truncate the above sums over k at k ≤ K ′ at the cost of O q g−1 K ′ +1 −2g /g . A 2 (Φ, g).…”

Section: Now We Use Lemma 34 For the Sums Over H And Getmentioning

confidence: 93%

Zeros of quadratic Dirichlet $L$-functions in the hyperelliptic ensemble

Bui

Florea

2018

Trans. Amer. Math. Soc.

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We study the 1-level density and the pair correlation of zeros of quadratic Dirichlet L-functions in function fields, as we average over the ensemble H 2g+1 of monic, square-free polynomials with coefficients in F q [x]. In the case of the 1-level density, when the Fourier transform of the test function is supported in the restricted interval ( 1 3 , 1), we compute a secondary term of size q − 4g 3 /g, which is not predicted by the Ratios Conjecture. Moreover, when the support is even more restricted, we obtain several lower order terms. For example, if the Fourier transform is supported in ( 1 3 , 1 2 ), we identify another lower order term of size q − 8g 5 /g. We also compute the pair correlation, and as for the 1-level density, we detect lower order terms under certain restrictions; for example, we see a term of size q −g /g 2 when the Fourier transform is supported in ( 1 4 , 1 2 ). The 1-level density and the pair correlation allow us to obtain non-vanishing results for L( 1 2 , χ D ), as well as lower bounds for the proportion of simple zeros of this family of L-functions.

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Section: Now We Use Lemma 34 For the Sums Over H And Getmentioning

confidence: 93%

Zeros of quadratic Dirichlet $L$-functions in the hyperelliptic ensemble

Bui

Florea

2018

Trans. Amer. Math. Soc.

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“…In this connection, we note that a lower order term involving is also present in the -level density of the family of Dirichlet -functions attached to all characters modulo (see [FM15, Theorem 1.2]). However, in this family this term is of order and is thus much smaller than in the family of real characters.…”

Section: Introductionmentioning

confidence: 99%

Low-lying zeros of quadratic Dirichlet -functions: lower order terms for extended support

Fiorilli¹,

Parks

Södergren

2017

Compositio Math.

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Abstract. We study the 1-level density of low-lying zeros of Dirichlet L-functions attached to real primitive characters of conductor at most X. Under the Generalized Riemann Hypothesis, we give an asymptotic expansion of this quantity in descending powers of log X, which is valid when the support of the Fourier transform of the corresponding even test function φ is contained in (−2, 2). We uncover a phase transition when the supremum σ of the support of φ reaches 1, both in the main term and in the lower order terms. A new lower order term appearing at σ = 1 involves the quantity φ(1), and is analogous to a lower order term which was isolated by Rudnick in the function field case.

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“…3 Allowing repetitions in the family in order to make the analysis manageable is not a new strategy; cf., e.g., [Y2,FM].…”

Section: Introductionmentioning

confidence: 99%

Low-lying zeros of elliptic curve L-functions: Beyond the Ratios Conjecture

Fiorilli¹,

Parks²,

Södergren³

2016

Math. Proc. Camb. Phil. Soc.

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Abstract. We study the low-lying zeros of L-functions attached to quadratic twists of a given elliptic curve E defined over Q. We are primarily interested in the family of all twists coprime to the conductor of E and compute a very precise expression for the corresponding 1-level density. In particular, for test functions whose Fourier transforms have sufficiently restricted support, we are able to compute the 1-level density up to an error term that is significantly sharper than the square-root error term predicted by the L-functions Ratios Conjecture.

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Surpassing the ratios conjecture in the 1-level density of DirichletL-functions

Cited by 25 publications

References 63 publications

Zeros of quadratic Dirichlet $L$-functions in the hyperelliptic ensemble

Zeros of quadratic Dirichlet $L$-functions in the hyperelliptic ensemble

Low-lying zeros of quadratic Dirichlet -functions: lower order terms for extended support

Low-lying zeros of elliptic curve L-functions: Beyond the Ratios Conjecture

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