Small gaps in coefficients of $L$-functions and $\mathfrak{B}$-free numbers in short intervals

Kowalski, Emmanuel; Robert, Olivier; Wu, Jie

doi:10.4171/rmi/496

Cited by 36 publications

(48 citation statements)

References 35 publications

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“…It is worth noticing that the case p : f (p)<0 1 p < ∞ is easier than the general case -actually it follows from work of Kowalski, Robert and Wu [22] on B-free numbers in short intervals that f has a sign change in all intervals [x, x + x θ ] for any θ > 7/17.…”

Section: By (34)mentioning

confidence: 99%

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: By (34)mentioning

confidence: 99%

Multiplicative functions in short intervals

2016

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“…The latest results can be found in the paper [13] by Kowalski, Robert and Wu. In particular, they showed that the gap between consecutive B-free numbers of size x is O(x 7/17 ).…”

Section: Here X Denotes the Distance From X To The Nearest Integermentioning

confidence: 92%

“…by condition (ii) and (13). Writing f (x) = o η (|g(x)|) when, for any fixed η > 0, f (x)/|g(x)| → 0 when x → ∞, we see by (12), (ii) and (1) that…”

Section: Proof Of Propositionmentioning

confidence: 98%

“…In the case of all short intervals, Kowalski, Robert and Wu [13] have made a comprehensive study. While considering almost all intervals here, we are able to take advantage of Theorem 2(ii) thanks to them: They refined Serre's argument to yield…”

Section: Here X Denotes the Distance From X To The Nearest Integermentioning

confidence: 99%

“…In [13], A 2 and A 3 were treated as one sum, but in doing that there is a slight problem due to definitions of two parameters η and l depending on each other. This problem is easy to overcome by working as here, that is as in [20].…”

Section: Proof Of Propositionmentioning

confidence: 99%

See 2 more Smart Citations

On the Distribution of -Free Numbers and Non-Vanishing Fourier Coefficients of Cusp Forms

Matomäki¹

2012

Glasgow Math. J.

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Abstract. We study properties of B-free numbers, that is numbers that are not divisible by any member of a set B. First we formulate the most-used procedure for finding them (in a given set of integers) as easy-to-apply propositions. Then we use the propositions to consider Diophantine properties of B-free numbers and their distribution on almost all short intervals. Results on B-free numbers have implications to non-vanishing Fourier coefficients of cusp forms, so this work also gives information about them.2010 Mathematics Subject Classification. 11F30, 11J25, 11N25.

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Exponential Sums

Bordellès¹

2012

Universitext

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Small gaps in coefficients of $L$-functions and $\mathfrak{B}$-free numbers in short intervals

Abstract: We discuss questions related to the non-existence of gaps in the series defining modular forms and other arithmetic functions of various types, and improve results of Serre, Balog & Ono and Alkan using new results about exponential sums and the distribution of B-free numbers.

Cited by 36 publications

References 35 publications

Multiplicative functions in short intervals

Multiplicative functions in short intervals

On the Distribution of -Free Numbers and Non-Vanishing Fourier Coefficients of Cusp Forms

Exponential Sums

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