Power mean values of the Riemann zeta‐function

Deshouillérs, Jean-Marc; Iwaniec, Henryk

doi:10.1112/s0025579300012298

Cited by 34 publications

(41 citation statements)

References 5 publications

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“…This echoes the way in which the results [5, Theorems 10 and 11], which bound sums involving the classical analogue of the Kloosterman sum S(r * ω, ω ′ ; ks) in (1.4.4), were used by Deshouillers and Iwaniec to obtain, in [6], new upper bounds for the mean value…”

Section: New Results On Sums Over Exceptional Eigenvaluesmentioning

confidence: 74%

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Weighted spectral large sieve inequalities for Hecke congruence subgroups of $SL(2,\mathbb{Z}[i])$

Watt¹

2013

Funct. Approx. Comment. Math.

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We prove new bounds for weighted mean values of sums involving Fourier coefficients of cusp forms that are automorphic with respect to a Hecke congruence subgroup Γ ≤ SL(2, Z[i]), and correspond to exceptional eigenvalues of the Laplace operator on the space L 2 (Γ\SL(2, C)/SU (2)). These results are, for certain applications, an effective substitute for the generalised Selberg eigenvalue conjecture. We give a proof of one such application, which is an upper bound for a sum of generalised Kloosterman sums (of significance in the study of certain mean values of Hecke zeta-functions with groessencharakters).Our proofs make extensive use of Lokvenec-Guleska's generalisation of the Bruggeman-Motohashi summation formulae for P SL(2, Z[i])\P SL(2, C). We also employ a bound of Kim and Shahidi for the first eigenvalues of the relevant Laplace operators, and an 'unweighted' spectral large sieve inequality (our proof of which is to appear separately).

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Section: New Results On Sums Over Exceptional Eigenvaluesmentioning

confidence: 74%

“…Given the above hypotheses, and given that (1.3.6) and (5.9) imply the bounds 6) it is trivially the case that…”

Section: By This Bound and That In (547) The Proof Of The Results (mentioning

confidence: 99%

Weighted spectral large sieve inequalities for Hecke congruence subgroups of $SL(2,\mathbb{Z}[i])$

Watt¹

2013

Funct. Approx. Comment. Math.

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“…Later, Conrey [2] generalized Levinson's argument (see also Iwaniec's book [6]) and proved that at least two-fifths of the zeros of ζ (s) are on the critical line. The crucial point of his paper is an elegant application of the upper bound for a certain Kloosterman sum by Deshouillers and Iwaniec (see [3,4]), which enabled him to compute the moment of the zeta-function twisted by a Dirichlet polynomial longer than Levinson's. To describe their results in detail, we prepare several notations.…”

Section: Introductionmentioning

confidence: 99%

An Application of Generalized Mollifiers to the Riemann Zeta-Function

Sono

2018

Kyushu J. Math.

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“…A mean value result where the Dirichlet polynomials are exactly of the form corresponding to our sum has been proved by Deshouillers and Iwaniec [8]. However, as we are working on very short intervals, we cannot afford to transform into Dirichlet polynomials and use their result.…”

Section: Proof Of Theorem 2 We Prove Theorem 2 By Showing That the Cmentioning

confidence: 99%

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

Matomäki¹

2012

Glasgow Math. J.

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Abstract. We study properties of B-free numbers, that is numbers that are not divisible by any member of a set B. First we formulate the most-used procedure for finding them (in a given set of integers) as easy-to-apply propositions. Then we use the propositions to consider Diophantine properties of B-free numbers and their distribution on almost all short intervals. Results on B-free numbers have implications to non-vanishing Fourier coefficients of cusp forms, so this work also gives information about them.2010 Mathematics Subject Classification. 11F30, 11J25, 11N25.

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Power mean values of the Riemann zeta‐function

Cited by 34 publications

References 5 publications

Weighted spectral large sieve inequalities for Hecke congruence subgroups of $SL(2,\mathbb{Z}[i])$

Weighted spectral large sieve inequalities for Hecke congruence subgroups of $SL(2,\mathbb{Z}[i])$

An Application of Generalized Mollifiers to the Riemann Zeta-Function

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

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