A New Lower Bound for the <i>L</i>
                        <sup>1</sup>
 Mean of the Exponential Sum with the Möbius Function

Balog, Antal; Ruzsa, Imre Z.

doi:10.1112/s0024609398005414

Cited by 14 publications

(15 citation statements)

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“…The earlier result on these exponential sums appears to be n≤x µ(n)e i2παn = O(x(log x) −c ), c > 0, in [22]. Recent contributions appear in [21], [68], [48], [9], [5], [63], [64], et alii.…”

Section: Exponential Sums Estimatesmentioning

confidence: 99%

Note on the Theory of Correlation Functions

Carella¹

2016

Preprint

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The correlation functions of some arithmetic functions are investigated in this note. The upper bounds for the autocorrelation of the Mobius function of degrees two and three, where C > 0 is a constant, respectively, are computed using elementary methods. These results are summarized as a 2-level autocorrelation function. Furthermore, some asymptotic result for the autocorrelation of the vonMangoldt function of degrees two n≤x Λ(n)Λ(n + 2t) ≫ x, where t ∈ Z, are computed using elementary methods. These results are summarized as a 3-level autocorrelation function.

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Section: Exponential Sums Estimatesmentioning

confidence: 99%

Note on the Theory of Correlation Functions

Carella¹

2016

Preprint

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“…We begin by applying Lemma 2•1 to get a test function for F θ(A i j ) for all pairs of indices {i, j} for which A i j is non-empty. Let b (1) i j , b (2) i j , . .…”

Section: Multidimensional Sets In Zmentioning

confidence: 99%

The L¹-norm of exponential sums in ^d

Petridis¹

2013

Math. Proc. Camb. Phil. Soc.

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Let A be a finite set of integers and F A (x) = a∈A exp(2πiax) be its exponential sum. McGehee, Pigno and Smith and Konyagin have independently proved that F A 1 c log |A| for some absolute constant c. The lower bound has the correct order of magnitude and was first conjectured by Littlewood. In this paper we present lower bounds on the L 1 -norm of exponential sums of sets in the d-dimensional grid Z d . We show that F A 1 is considerably larger than log |A| when A ⊂ Z d has multidimensional structure. We furthermore prove similar lower bounds for sets in Z, which in a technical sense are multidimensional and discuss their connection to an inverse result on the theorem of McGehee, Pigno and Smith and Konyagin.

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“…Будем обозначать BRJV(5O) = BRjy-А. Балог и И. Ружи [1] фактически дока зали, что BRiv <C N 3/4 log TV, N ^ 2.…”

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