Sign changes of Hecke eigenvalues

Matomäki, Kaisa; Radziwiłł, Maksym

doi:10.1007/s00039-015-0350-7

Cited by 29 publications

(34 citation statements)

References 21 publications

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“…In this case Corollary 3 recovers a recent result of the authors [30]. As observed by Ghosh and Sarnak in [10], the number of sign changes of λ f (n) for n ≤ k 1/2 (with k the weight of f ) is related to the number of zeros of f on the vertical geodesic high in the cusp.…”

Section: Corollarysupporting

confidence: 86%

“…Using Halász's theorem one can show that if f (p)<0 1/p = ∞ and f (n) = 0 for a positive proportion of the integers n then the non-zero values of f (n) are half of the time positive and half of the time negative (see [30,Lemma 2.4] or [7,Lemma 3.3]). Since we expect f (n) and f (n + 1) to behave independently this suggests that, for non-vanishing f such that f (p)<0 1/p = ∞, there should be about x/2 sign changes among integers n ≤ x.…”

Section: Corollarymentioning

confidence: 99%

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Multiplicative functions in short intervals

2016

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Abstract. We introduce a general result relating "short averages" of a multiplicative function to "long averages" which are well understood. This result has several consequences. First, for the Möbius function we show that there are cancellations in the sum of µ(n) in almost all intervals of the form [x, x + ψ(x)] with ψ(x) → ∞ arbitrarily slowly. This goes beyond what was previously known conditionally on the Density Hypothesis or the stronger Riemann Hypothesis. Second, we settle the long-standing conjecture on the existence of x ǫ -smooth numbers in intervals of the form [x, x + c(ε) √ x], recovering unconditionally a conditional (on the Riemann Hypothesis) result of Soundararajan. Third, we show that the mean-value of λ(n)λ(n + 1), with λ(n) Liouville's function, is non-trivially bounded in absolute value by 1−δ for some δ > 0. This settles an old folklore conjecture and constitutes progress towards Chowla's conjecture. Fourth, we show that a (general) real-valued multiplicative function f has a positive proportion of sign changes if and only if f is negative on at least one integer and non-zero on a positive proportion of the integers. This improves on many previous works, and is new already in the case of the Möbius function. We also obtain some additional results on smooth numbers in almost all intervals, and sign changes of multiplicative functions in all intervals of square-root length.

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Section: Corollarysupporting

confidence: 86%

Section: Corollarymentioning

confidence: 99%

Multiplicative functions in short intervals

2016

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“…For general coefficients of holomorphic cusp forms one can prove Assumption A1 for all primes with at most finitely many exceptions by combining [19,Lemma 9] and [18, Lemma 2.2]. For coefficients of elliptic curves Assumption A2 can be proven for fixed α, β, on average for most elliptic curves (see [1,24,25] for the currently strongest forms of this statement).…”

Section: Resultsmentioning

confidence: 99%

On the typical size and cancellations among the coefficients of some modular forms

Luca¹,

Radziwiłł²,

Shparlinski³

2018

Math. Proc. Camb. Phil. Soc.

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We obtain a nontrivial upper bound for almost all elements of the sequences of real numbers which are multiplicative and at the prime indices are distributed according to the Sato-Tate density. Examples of such sequences come from coefficients of several L-functions of elliptic curves and modular forms. In particular, we show that |τ (n)| ≤ n 11/2 (log n) −1/2+o(1) for a set of n of asymptotic density 1, where τ (n) is the Ramanujan τ function while the standard argument yields log 2 instead of −1/2 in the power of the logarithm. Another consequence of our result is that in the number of representations of n by a binary quadratic form one has slightly more than square-root cancellations for almost all integers n.In addition we obtain a central limit theorem for such sequences, assuming a weak hypothesis on the rate of convergence to the Sato-Tate law. For Fourier coefficients of primitive holomorphic cusp forms such a hypothesis is known conditionally assuming the automorphy of all symmetric powers of the form and seems to be within reach unconditionally using the currently established potential automorphy.

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“…By using the method of [17] based on multiplicative function theory, we establish the following result. Theorem 2.…”

Section: Introductionmentioning

confidence: 99%

Non-vanishing and sign changes of Hecke eigenvalues for half-integral weight cusp forms

Chen

2016

Indagationes Mathematicae

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In this paper, we consider three problems about signs of the Fourier coefficients of a cusp form f with half-integral weight:-The first negative coefficient of the sequence {a f (tn 2 )} n∈N , -The number of coefficients a f (tn 2 ) of same signs, -Non-vanishing of coefficients a f (tn 2 ) in short intervals and in arithmetic progressions, where a f (n) is the n-th Fourier coefficient of f and t is a square-free integer such that a f (t) = 0.

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Sign changes of Hecke eigenvalues

Cited by 29 publications

References 21 publications

Multiplicative functions in short intervals

Multiplicative functions in short intervals

On the typical size and cancellations among the coefficients of some modular forms

Non-vanishing and sign changes of Hecke eigenvalues for half-integral weight cusp forms

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