An arithmetical mapping and applications to Ω-results for the Riemann zeta function

Hilberdink, Titus

doi:10.4064/aa139-4-3

Cited by 49 publications

(93 citation statements)

References 12 publications

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“…, c N ) in C N . Following [1, Section 4], we obtain [9], as mentioned in Section 1. (2) Note that the GCD sums seem indispensable for estimating Λ t (N ) because of the central role played by the completeness property.…”

Section: Discussionmentioning

confidence: 99%

GCD sums and complete sets of square-free numbers

Bondarenko

Seip

2014

Bulletin of the London Mathematical Society

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ABSTRACT. It is proved that≪ N exp C log N log log log N log log N holds for arbitrary integers 1 ≤ n 1 < · · · < n N . This bound is essentially better than that found in a recent paper of Aistleitner, Berkes, and Seip and can be improved by no more than removal of the triple logarithm. A certain completeness property of extremal sets of square-free numbers plays an important role in the proof of this result.

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Section: Discussionmentioning

confidence: 99%

GCD sums and complete sets of square-free numbers

Bondarenko

Seip

2014

Bulletin of the London Mathematical Society

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“…by Mertens' third theorem with an explicit estimate for the error term from [27,Theorem 8]. For the other product in (16)…”

Section: Proof Of Theoremmentioning

confidence: 99%

On large values of L(σ,χ)

Aistleitner

Mahatab

Munsch

et al. 2018

The Quarterly Journal of Mathematics

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In recent years a variant of the resonance method was developed which allowed to obtain improved Ω-results for the Riemann zeta function along vertical lines in the critical strip. In the present paper we show how this method can be adapted to prove the existence of large values of |L(σ, χ)| in the range σ ∈ (1/2, 1], and to estimate the proportion of characters for which |L(σ, χ)| is of such a large order. More precisely, for every fixed σ ∈ (1/2, 1) we show that for all sufficiently large q there is a non-principal character χ (mod q) such that log |L(σ, χ)| ≥ C(σ)(log q) 1−σ (log log q) −σ . In the case σ = 1 we show that there is a non-principal character χ (mod q) for which |L(1, χ)| ≥ e γ (log 2 q + log 3 q − C). In both cases, our results essentially match the prediction for the actual order of such extreme values, based on probabilistic models.

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“…Dans le prolongement des travaux de Montgomery et Balasubramanian–Ramachandra , Bondarenko et Seip ont récemment obtenu — voir notamment [, ] — des bornes inférieures pour la quantité

Z_{β} false(T false) : = \underset{T^{β} ⩽ τ ⩽ T}{trueprefixmax} |ζ false(0false \frac{1}{2} + i τ false)| false(0 ⩽ β < 1, T ⩾ 1 false)

en minorant la norme de la forme quadratique associée à la sous‐somme

S (M)

. Leur approche repose sur la méthode de résonance, développée indépendamment, dans ce cadre, par Soundararajan et Hilberdink . Elle est adaptable à l'étude des maximums d'autres séries de Dirichlet — voir la publication récente .…”

Section: Introduction Et éNoncé Des Résultatsunclassified

“…Dans , ce résultat est étendu à

c < \sqrt{1 - β}

ce qui représente une amélioration d'un facteur

\sqrt{2 false(1 - β false)}

lorsque

β ⩽ \frac{1}{2}

. Le Théorème améliore donc les exposants obtenus par Bondarenko et Seip d'un nouveau facteur

\sqrt{2}

pour tout

β \in [0, 1 [

. (ii)Notre approche diffère sensiblement de celle de : à l'instar des méthodes développées dans [, ], nous relions le carré du maximum à la somme de Gál complète

S (M)

. L'introduction du carré explique que le gain obtenu n'est que la moitié de celui qui a été obtenu pour la somme de Gál soit

\frac{1}{2} 2 \sqrt{2} = \sqrt{2}

.…”

Section: Introduction Et éNoncé Des Résultatsunclassified

Sommes de Gál et applications

Bretèche

Tenenbaum

2018

Proc. London Math. Soc.

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We evaluate the asymptotic size of various sums of Gál type, in particular 0trueS(M):=∑m,n∈scriptM(m,n)[m,n],where scriptM is a finite set of integers. Elaborating on methods recently developed by Bondarenko and Seip, we obtain an asymptotic formula for log supfalse|scriptMfalse|=NS(M)/Nand derive new lower bounds for localized extreme values of the Riemann zeta‐function, for extremal values of some Dirichlet L‐functions at s=12, and for large character sums.

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An arithmetical mapping and applications to Ω-results for the Riemann zeta function

Cited by 49 publications

References 12 publications

GCD sums and complete sets of square-free numbers

GCD sums and complete sets of square-free numbers

On large values of L(σ,χ)

Sommes de Gál et applications

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