Some mean value results for the Riemann zeta-function

Ivić, Aleksandar

doi:10.1515/9783110870923.145

Cited by 49 publications

(102 citation statements)

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“…In order to obtain the bound (8.9), we note firstly that, by (8.7) and Leibniz's rule for higher order derivatives of products, one has: 10) for some non-negative integers ℓ, m with ℓ ≤ j and m ≤ k. Then we observe that, by (5.19), one has…”

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

confidence: 98%

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

confidence: 98%

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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“…in Section 2.1 of [9]. More precisely, the claim follows from Lemma 2.1 of [9] and integration by parts.…”

Section: Lemmas On Exponential Integralsmentioning

confidence: 84%

“…For comparison, during the 20th century the error term in the Dirichlet divisor problem saw a long string of upper bounds improving the cubic root cancellation first obtained by Voronoi [28]. For more on the rather extensive literature on the Dirichlet divisor problem, we recommend Tsang's survey [27] and references therein, and Chapter 13 in the book [9] of Ivić.…”

Section: Linear Exponential Sums With Rational Twistsmentioning

confidence: 99%

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Moments and oscillations of exponential sums related to cusp forms

Vesalainen

2016

Math. Proc. Camb. Phil. Soc.

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We consider large values of long linear exponential sums involving Fourier coefficients of holomorphic cusp forms. The sums we consider involve rational linear twists e(nh/k) with sufficiently small denominators. We prove both pointwise upper bounds and bounds for the frequency of large values. In particular, the k-aspect is treated. As an application we obtain upper bounds for all the moments of the sums in question. We also give the asymptotics with the right main term for fourth moments.We also consider the mean square of very short sums, proving that on average short linear sums with rational additive twists exhibit square root cancellation. This result is also proved in a slightly sharper form.Finally, the consideration of moment estimates for both long and short exponential sums culminates in a result concerning the oscillation of the long linear sums. Essentially, this result says that for a positive proportion of time, such a sum stays in fairly long intervals, where its order of magnitude does not drop below the average order of magnitude and where its argument is in a given interval of length 3π/2 + ε and so can not wind around the origin.

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“…Would it be possible to balance the techno-scientific postmodern evolution and the thinking of the creative human being? Probably this issue will has to stay open even if, as it is done in this note, we try to give a scientific answer to this question by using the image of zeta function properties [11][12][13][14][15]. Indeed we think that this mathematical image could be a key of understanding the world of complex systems.…”

Section: Introductionmentioning

confidence: 99%

Role of Riemann's and Goldbach's hypotheses in the behaviour of complex systems:Introduction to the concept of "Sciances"

Méhauté

Tayurskiı̆

2012

J. Phys.: Conf. Ser.

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Abstract. The authors have already established a bi univocal correspondence betweenRiemann zeta functions and dynamic processes under the control of integro-differential operator of non-integer complex order. We recall that the Riemann zeta function can then be related to hyperbolic geodesics whose angles at the boundary are determined by the real part of the power laws that define the Riemann series. It is suggested that Riemann's conjecture can be reduced to a geometrical phase transition with a reduction of the parameter of order resulting from the combination of a pair symmetries associated with a quasi-self similarity of geodetics. The well-known relationship with the set of prime numbers must be considered as the result of the local existence of stationary 'state' in the dynamics. The work is focused on the 'non stationary' behaviour of Riemann zeta function. It is shown that the main characteristic of the dynamics of complex systems may be associated to a hybridizing between a pair of states and/or processes able to give a geometrical status to the concept of the time and equally to the concept of energy. It is based on the mirror properties of complementary zeta function. It is shown also that the set of prime numbers, which controls the transitions between 'states', is the simplest form of the complexity. This analysis suggests the existence of a mathematical relationship between Riemann's and Goldbach's hypothesis. Such relationship would be the base of an extension of the principles of the science for the analysis of the complexity. According to previous proposal we name this extension that enlarges the principles of the science: sciance with 'a'.

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Some mean value results for the Riemann zeta-function

Cited by 49 publications

References 0 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 $SL(2,\mathbb{Z}[i])$

Moments and oscillations of exponential sums related to cusp forms

Role of Riemann's and Goldbach's hypotheses in the behaviour of complex systems:Introduction to the concept of "Sciances"

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