The Wiener–Ikehara theorem by complex analysis

Korevaar, J.

doi:10.1090/s0002-9939-05-08060-3

Cited by 9 publications

(3 citation statements)

References 20 publications

(18 reference statements)

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“…We now use the following version of the famous Wiener-Ikehara theorem taken from [8] in order to conclude the existence of natural density instead of 'only' Dedekind-Dirichlet density in some cases.…”

Section: Towards Natural Densitymentioning

confidence: 99%

On conjectures of Sato–Tate and Bruinier–Kohnen

2014

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This article covers three topics. (1) It establishes links between the density of certain subsets of the set of primes and related subsets of the set of natural numbers. (2) It extends previous results on a conjecture of Bruinier and Kohnen in three ways: the CM-case is included; under the assumption of the same error term as in previous work one obtains the result in terms of natural density instead of Dedekind-Dirichlet density; the latter type of density can already be achieved by an error term like in the prime number theorem. (3) It also provides a complete proof of Sato-Tate equidistribution for CM modular forms with an error term similar to that in the prime number theorem.In this section we are concerned with the setsfor a multiplicative arithmetic function χ : N → {−1, 0, +1}, as explained in the introduction. We found it necessary to assume more than just that S ± has a natural density in order to conclude something about the density of A ± ; namely, we obtain our results under the assumption that S ± is (weakly) regular (see below). We also show that (weak) regularity is a consequence of a sufficiently good error bound for the convergence of the natural density of S ± . P <0 = {p ∈ P : ψ(p) = −1}, and P =0 = {p ∈ P : ψ(p) = 0}. Theorem 4.1.1 shows that these sets are weakly regular and allows us to conclude due to Proposition 2.3.1. Corollary 4.3.2. Let χ as above. Assume the setting of part (d) of Theorem 4.1.1. Then the sets {n ∈ N | n ∈ N χ and a(tn 2 ) > 0} and {n ∈ N | n ∈ N χ and a(tn 2 ) < 0} have equal positive natural densities, that is, both are precisely half of the density of the set {n ∈ N | n ∈ N χ and a(tn 2 ) = 0}.Proof. The proof proceeds precisely as that of Corollary 4.3.1, except that in the end we appeal to Proposition 2.5.2.

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Section: Towards Natural Densitymentioning

confidence: 99%

On conjectures of Sato–Tate and Bruinier–Kohnen

2014

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“…Newman's method was later adapted to other Tauberian problems in numerous articles, see e.g. [1,2,23,27,28,43] and the various bibliographical remarks in [25,Chap. III].…”

Section: Introductionmentioning

confidence: 99%

Complex Tauberian theorems for Laplace transforms with local pseudofunction boundary behavior

Debruyne

Vindas

2019

JAMA

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We provide several Tauberian theorems for Laplace transforms with local pseudofunction boundary behavior. Our results generalize and improve various known versions of the Ingham-Fatou-Riesz theorem and the Wiener-Ikehara theorem. Using local pseudofunction boundary behavior enables us to relax boundary requirements to a minimum. Furthermore, we allow possible null sets of boundary singularities and remove unnecessary uniformity conditions occurring in earlier works; to this end, we obtain a useful characterization of local pseudofunctions. Most of our results are proved under one-sided Tauberian hypotheses; in this context, we also establish new boundedness theorems for Laplace transforms with pseudomeasure boundary behavior. As an application, we refine various results related to the Katznelson-Tzafriri theorem for power series.

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“…where b ∈ C and k is holomorphic on Re(z) ≥ 1. We can now apply Wiener-Ikehara's theorem (see [8]) to deduce the result.…”

Section: Lemmamentioning

confidence: 97%

The equidistribution of Fourier coefficients of half integral weight modular forms on the plane

Mezroui¹

2019

Czech.Math.J.

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Let f = ∞ n=1 a(n)q n ∈ S k+1/2 (N, χ 0 ) be a non-zero cuspidal Hecke eigenform of weight k + 1 2 and the trivial nebentypus χ 0 where the Fourier coefficients a(n) are real. Bruinier and Kohnen conjectured that the signs of a(n) are equidistributed. This conjecture was proved to be true by Inam, Wiese and Arias-de-Reyna for the subfamilies {a(tn 2 )} n where t is a squarefree integer such that a(t) = 0. Let q and d be natural numbers such that (d, q) = 1. In this work, we show that {a(tn 2 )} n is equidistributed over any arithmetic progression n ≡ d mod q.

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The Wiener–Ikehara theorem by complex analysis

Cited by 9 publications

References 20 publications

On conjectures of Sato–Tate and Bruinier–Kohnen

On conjectures of Sato–Tate and Bruinier–Kohnen

Complex Tauberian theorems for Laplace transforms with local pseudofunction boundary behavior

The equidistribution of Fourier coefficients of half integral weight modular forms on the plane

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