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

Matomäki, Kaisa

doi:10.1017/s0017089512000043

Cited by 7 publications

(7 citation statements)

References 18 publications

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“…This corollary can be compared with the results of Matomäki [11]. In [11], N was allowed to grow with X and, provided that N ≤ X 1 6 −δ for some δ > 0, Matomäki showed that P B (X, N ) N −θ for all X and any θ < 1.…”

Section: Introductionmentioning

confidence: 89%

“…Erdös conjectured that for any constant θ > 0, for all sufficiently large x, the interval x, x + x θ must contain a B-free number, and proved existence of a θ < 1 which is independent of B and has this property. Since then, B-free integers in short intervals and, consequently, the gaps between them have been extensively studied (e.g., see [10,11,14,18]).…”

Section: Introductionmentioning

confidence: 99%

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Variance of $\mathcal{B}$-free integers in short intervals

Avdeeva¹

2015

Preprint

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We prove some new statements on the distribution of Bfree numbers in short intervals. In particular, we show an asymptotic result for the variance of the number of B-free integers in random short intervals which are, in some sense, uniformly distributed.We establish a connection between our work and the paper by El Abdalaoui, Lemańczyk & de la Rue on a flow associated to B-free integers. In addition, we study an analog of our variance for k-free integers in a number field which provides new information for the corresponding dynamical system constructed by Cellarosi & Vinogradov. † As usually, here the subscript in the O-notation emphasizes dependence on B.

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Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Variance of $\mathcal{B}$-free integers in short intervals

Avdeeva¹

2015

Preprint

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“…In Section 6.2 we give a further application of Theorem 1.9 to estimates for the frequency of long gaps between consecutive B-free numbers, improving results of Plaksin [37] and Matomäki [27].…”

Section: We Denote Bymentioning

confidence: 80%

“…Suppose B is a regularly varying sequence of index α ∈ (0, 1) and ϕ is a real-valued function of bounded variation and supported in a compact subset of [0, ∞) and non-vanishing on some open interval. Then the random variable Improving on work of Plaksin [37], Matomäki [27] used a sieve-theoretic method to show that for any ε > 0, |G(X, H)| XH −1+ε for 1 ≤ H ≤ X 1/6−ε (with no upper bound constraint on the range of H if B consists only of primes). As a consequence of our kth-moment bounds, we can prove the following.…”

Section: 2mentioning

confidence: 99%

Squarefrees are Gaussian in short intervals

Gorodetsky,

Mangerel,

Rodgers

2021

Preprint

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We show that counts of squarefree integers up to X in short intervals of size H tend to a Gaussian distribution as long as H → ∞ and H = X o(1) . This answers a question posed by R.R. Hall in 1989. More generally we prove a variant of Donsker's theorem, showing that these counts scale to fractional Brownian motion with Hurst parameter 1/4. In fact we are able to prove these results hold in general for collections of B-free integers as long as the sieving set B satisfies a very mild regularity property, for Hurst parameter varying with the set B.

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“…This ends our digression on the uniformity of q in an almost all sense. For research on other types of denser sequences such as the B-free numbers and application to the size of gaps between consecutive nonzero Fourier coefficients of modular forms via sieve methods, see [1,2,13].…”

Section: Introductionmentioning

confidence: 99%

Number of prime ideals in short intervals

Alkan

Mehreliyev

2016

Journal of Number Theory

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Primes in a progression Dedekind zeta function Dirichlet L-function Branch of complex logarithm Linear forms in logarithms Siegel zeroAssuming a weaker form of the Riemann hypothesis for Dedekind zeta functions by allowing Siegel zeros, we extend a classical result of Cramér on the number of primes in short intervals to prime ideals of the ring of integers in cyclotomic extensions with norms belonging to such intervals. The extension is uniform with respect to the degree of the cyclotomic extension. Our approach is based on the arithmetic of cyclotomic fields and analytic properties of their Dedekind zeta functions together with a lower bound for the number of primes over progressions in short intervals subject to similar assumptions. Uniformity with respect to the modulus of the progression is obtained and the lower bound turns out to be best possible, apart from constants, as shown by the BrunTitchmarsh theorem.

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

Cited by 7 publications

References 18 publications

Variance of $\mathcal{B}$-free integers in short intervals

Variance of $\mathcal{B}$-free integers in short intervals

Squarefrees are Gaussian in short intervals

Number of prime ideals in short intervals

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