Mean values with cubic characters

Baier, Stephan; Young, Matthew P.

doi:10.1016/j.jnt.2009.11.007

Cited by 60 publications

(105 citation statements)

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“…This generalization of the classical large sieve inequality to sparse moduli sets, usually given as values of some fixed polynomial, has been studied intensely in past research and has already influenced some other areas of Number Theory, especially the version with k = 2 from [2]. To name a few such topics, it has been found useful to find variants of Bombieri-Vinogradov's theorem [4,13], new results on primes of polynomial shape [9] or primes in APs to spaced moduli [4], divisibility questions with Fermat quotients [8], and mean value estimates for character sums with applications [1,3,5,16]. Furthermore, the multidimensional polynomial LSI can be used for sieving with high powers like seen in [12] and reaches questions related to the abc-conjecture.…”

Section: Second Application: the Polynomial Large Sieve Inequality (Lmentioning

confidence: 99%

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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Section: Second Application: the Polynomial Large Sieve Inequality (Lmentioning

confidence: 99%

Bounds for discrete moments of Weyl sums and applications

Halupczok

2020

Acta Arith.

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“…where g is a smooth compactly supported function arising from the partition of unity and the support of g is in the interval [1,2]. As noted in [28], it suffices to show that…”

Section: General Approachmentioning

confidence: 99%

On the low-lying zeros of Hasse–Weil L-functions for elliptic curves

Baier

Zhao

2008

Advances in Mathematics

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In this paper, we obtain an unconditional density theorem concerning the low-lying zeros of Hasse-Weil L-functions for a family of elliptic curves. From this together with the Riemann hypothesis for these L-functions, we infer the majorant of 27/14 (which is strictly less than 2) for the average rank of the elliptic curves in the family under consideration. This upper bound for the average rank enables us to deduce that, under the same assumption, a positive proportion of elliptic curves have algebraic ranks equaling their analytic ranks and finite Tate-Shafarevic group. Statements of this flavor were known previously [28] under the additional assumptions of GRH for Dirichlet L-functions and symmetric square L-functions which are removed in the present paper.

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“…Soundararajan [15] has shown that at least 7/8 of the central values in the family of quadratic Dirichlet L-functions are nonzero. More recently, Baier and Young [2] consider the family of Dirichlet L-functions associated to cubic and sextic characters and show that infinitely many (though not a positive proportion) of these functions are not zero at the central point.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%