Nair–Tenenbaum bounds uniform with respect to the discriminant

Henriot, Kevin

doi:10.1017/s0305004111000752

Cited by 47 publications

(70 citation statements)

References 16 publications

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“…Note that such a result could also be deduced from earlier work of Barban and Vekhov ; see also for the most recent results on this topic.…”

Section: Lemmassupporting

confidence: 79%

Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method

Drappeau

2017

Proc. London Math. Soc.

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International audienceWe prove a bound for quintilinear sums of Kloosterman sums, with congruence conditions on the "smooth" summation variables. This generalizes classical work of Deshouillers and Iwaniec, and is key to obtaining power-saving error terms in applications, notably the dispersion method. As a consequence, assuming the Riemann hypothesis for Dirichlet $L$-functions, we prove a power-saving error term in the Titchmarsh divisor problem of estimating $\sum_{p\leq x}\tau(p-1)$. Unconditionally, we isolate the possible contribution of Siegel zeroes, showing it is always negative. Extending work of Fouvry and Tenenbaum, we obtain power-saving in the asymptotic formula for $\sum_{n\leq x}\tau_k(n)\tau(n+1)$, reproving a result announced by Bykovski\u{i} and Vinogradov by a different method. The gain in the exponent is shown to be independent of $k$ if a generalized Lindel\"of hypothesis is assumed

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“…Note that such a result could also be deduced from earlier work of Barban and Vekhov ; see also for the most recent results on this topic.…”

Section: Lemmassupporting

confidence: 79%

Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method

Drappeau

2017

Proc. London Math. Soc.

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“…by a result of Henriot [9,10], since 1 λ f (n) =0 is a multiplicative function of n. In addition it follows from this that |U 2 | ≤ Cδ 4 X for some large absolute constant C > 0. Applying Cauchy-Schwarz and the previous two bounds, we get…”

Section: Theorem 11 Was Also Recently Obtained Independently (For Mamentioning

confidence: 74%

“…To obtain the lower bound (9) n≤x g(n)g(n + 1) ≥ (−1 + c)x, we first note that, by [5,Lemma 4.1], there is an even integer b ≪ f 1 such that g(2 b ) = 1. Notice that the claim is trivial unless g(2 j ) = 0 for all j ≤ b.…”

Section: Proof Of Corollary 13mentioning

confidence: 99%

Sign changes of Hecke eigenvalues

Matomäki

Radziwiłł

2015

Geom. Funct. Anal.

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“…Lorsque ξ/2 k > ξ ε/2 , nous sommes en mesure d'appliquer le théorème 4 de [21] pouŕ evaluer la sommation relativeà m ouà n, selon que α appartientà W k ou Z k . Il vient…”

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Sur la conjecture de Manin pour certaines surfaces de Châtelet

Bretèche¹,

Tenenbaum²

2013

J. Inst. Math. Jussieu

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International audienceWe prove Manin's conjecture, in the strong form conjectured by Peyre, for Châtelet surfaces associated to surfaces of the type y^2 + z^2 = P (x, 1), where P is a binary quartic form with integer coefficients that is either irreducible over Q[i] or the product of two quadratic forms with integer coefficients and irreducible over Q[i]. Moreover, we provide an explicit upper bound for the remainder term in the relevant asymptotic formula. This essentially settles Manin's conjecture for all Châtelet surfaces. The proof rests on two new tools, namely upper bounds for mean values of local oscillations of characters on divisors and sharp upper estimates for mean values of arithmetic functions of binary forms. Pezzo surfaces of degree 4, count of lattice points over algebraic varieties, number of representations as a sum of two squares, method of descent.Nous démontrons , sous la forme forte conjecturée par Peyre, la conjecture de Manin pour les surfaces de Châtelet dont les équations sont du type y^2 + z^2 = P (x, 1), où P est une forme binaire quartique à coefficients entiers irréductible sur Q[i] ou produit de deux formes quadratiques à coefficients entiers irréductibles sur Q[i]. De plus, nous fournissons une estimation explicite du terme d'erreur de la formule asymptotique sous-jacente. Cela finalise essentiellement la validation de la conjecture de Manin pour l'ensemble des surfaces de Châtelet. La preuve s'appuie sur deux méthodes nouvelles, concernant, du part, les estimations en moyenne d'oscillations locales de caractères sur les diviseurs, et, d'autre part, les majorations de certaines fonctions arithmétiques de formes binaires. Mots clefs : Surfaces de Châtelet , conjecture de Manin, constante de Peyre, surfaces de del Pezzo de degré 4, comptage de points entiers sur des variétés algébriques , nombre de représentations comme somme de deux carrés , méthode de descente

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Nair–Tenenbaum bounds uniform with respect to the discriminant

Abstract: For a suitable arithmetic function F and polynomials Q 1 , . . . , Q k in Z[X], Nair and Tenenbaum obtained an upper bound on the short sum x

Cited by 47 publications

References 16 publications

Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method

Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method

Sign changes of Hecke eigenvalues

Sur la conjecture de Manin pour certaines surfaces de Châtelet

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