Three-dimensional exponential sums with monomials

Robert, Olivier; Sargos, Patrick

doi:10.1515/crelle.2006.012

Cited by 112 publications

(84 citation statements)

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“…Lemma 1 was proved by Robert-Sargos [21]. It represents a powerful arithmetic tool which is essential in the analysis when the biquadrate of sums involving √ n appears in exponentials, and was used e.g., in [11].…”

Section: The Proof Of Theoremmentioning

confidence: 99%

On the divisor function and the Riemann zeta-function in short intervals

Ivić

2008

Ramanujan J

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Abstract. We obtain, for T ε ≤ U = U(T ) ≤ T 1/2−ε , asymptotic formulas forwhere ∆(x) is the error term in the classical divisor problem, and E(T ) is the error term in the mean square formula for |ζ( 1 2 + it)|. Upper bounds of the form O ε (T 1+ε U 2 ) for the above integrals with biquadrates instead of square are shown to hold for T 3/8 ≤ U = U(T ) ≪ T 1/2 . The connection between the moments of E(t + U) − E(t) and |ζ( 1 2 + it)| is also given. Generalizations to some other number-theoretic error terms are discussed. IntroductionPower moments represent one of the most important parts of the theory of the Riemann zeta-function ζ(s), defined asand otherwise by analytic continuation. Of particular significance are the moments on the "critical line" σ = 1 2 , and a vast literature exists on this subject (see e.g., the monographs [5], [6], and [23]). In this paper we shall be concerned with moments of the error function1991 Mathematics Subject Classification. 11 M 06, 11 N 37. Key words and phrases. The Riemann zeta-function, the divisor functions, power moments in short intervals, upper bounds.Typeset by A M S-T E X

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Section: The Proof Of Theoremmentioning

confidence: 99%

On the divisor function and the Riemann zeta-function in short intervals

Ivić

2008

Ramanujan J

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“…In this paper, combining the method of [7] and a recent deep result of Robert and Sargos [9], we shall prove the following…”

Section: (Nmkl) −3/4 D(n)d(m)d(k)d(l)mentioning

confidence: 80%

On higher-power moments of Δ(x) (III)

Zhai

2005

Acta Arith.

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“…Following the approach of Tsang [9], Zhai [10] proved that the equation ( Later, combining the method of [4] and a deep result of Robert and Sargos [8], Zhai [12] proved that the equation (1.4) holds for δ 4 = 3/28. By a unified approach, Zhai [11] proved that the asymptotic formula…”

Section: Introduction and Main Resultsmentioning

confidence: 99%