Arithmetic Tales

Bordellès, Olivier

doi:10.1007/978-3-030-54946-6

Cited by 14 publications

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“…Next, we need the following simple bound for the number of curve points close to integer points. This is Lemma 2 in [14], see also [6,Thm. 5.6], where a proof is provided.…”

Section: Introductionmentioning

confidence: 75%

Bounds for discrete moments of Weyl sums and applications

Halupczok

2020

Acta Arith.

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We prove two bounds for discrete moments of Weyl sums. The first one can be obtained using a standard approach. The second one involves an observation how this method can be improved, which leads to a sharper bound in certain ranges. The proofs both build on the recently proved main conjecture for Vinogradov's mean value theorem.We present two selected applications: First, we prove a new kth derivative test for the number of integer points close to a curve by an exponential sum approach. This yields a stronger bound than existing results obtained via geometric methods, but it is only applicable for specific functions. As second application we prove a new improvement of the polynomial large sieve inequality for one-variable polynomials of degree k ≥ 4.1991 Mathematics Subject Classification. Primary 11L15, Secondary 11J54.

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“…Next, we need the following simple bound for the number of curve points close to integer points. This is Lemma 2 in [14], see also [6,Thm. 5.6], where a proof is provided.…”

Section: Introductionmentioning

confidence: 75%

Bounds for discrete moments of Weyl sums and applications

Halupczok

2020

Acta Arith.

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“…Later, Kanemitsu and Sita Rama Chandra Rao [8] proved this conjecture in the case α 1 2 and j 2. They also study the mean square of G α,j (x) and proved the following bound which supports the Chowla-Walum Conjecture: if |α| 1 2 , then…”

Section: The Chowla-walum Conjecturementioning

confidence: 82%

On restricted divisor sums associated to the Chowla-Walum conjecture

Bordellès¹

2019

Preprint

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We first study the mean value of certain restricted divisor sums involving the Chowla-Walum sums, improving in particular a recent estimate given by Iannucci. The aim of the second part of this work is the generalization of the previous study, by restricting the range of the divisors in the studied divisor sums, extending the Chowla-Walum conjecture, proving a small part of this extended conjecture and generalizing the asymptotic formulas previously obtained in the first part.

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“…However, this generalization does not seem to bring added value and is left to the interested reader. Note that 0 ă Φptq ă 1 for 0 ă |t| ă 1.We follow the proof of[4, Corollary 6.2]. It follows from the result of Vaaler[20], which we use in the form given by [4, Theorem 6.1], that for any real number x ě 1 and any positive integer H, Using (4.1) with x " n{p `gppq, multiplying by plog pq κ and summing over all the primes p in pM, 2Ms yields the estimate…”

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confidence: 92%

Denominators of Bernoulli Polynomials

Bordellès¹,

Luca

Moree

et al. 2018

Mathematika

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For a positive integer n let P n " ź p sppnqěp p,where p runs over primes and s p pnq is the sum of the base p digits of n. For all n we prove that P n is divisible by all "small" primes with at most one exception. We also show that P n is large, has many prime factors exceeding ? n, with the largest one exceeding n 20{37 . We establish Kellner's conjecture, which says that the number of prime factors exceeding ? n grows asymptotically as κ ? n{ log n for some constant κ with κ " 2. Further, we compare the sizes of P n and P n`1 , leading to the somewhat surprising conclusion that although P n tends to infinity with n, the inequality P n ą P n`1 is more frequent than its reverse.

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Arithmetic Tales

Cited by 14 publications

References 0 publications

Bounds for discrete moments of Weyl sums and applications

Bounds for discrete moments of Weyl sums and applications

On restricted divisor sums associated to the Chowla-Walum conjecture

Denominators of Bernoulli Polynomials

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