Mellin vu du ciel Mellin, seen from the sky Philippe Flajolet - - PowerPoint PPT Presentation

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Mellin vu du ciel Mellin, seen from the sky Philippe Flajolet - - PowerPoint PPT Presentation

Mellin vu du ciel Mellin, seen from the sky Philippe Flajolet INRIA Rocquencourt March 10, 2008 1 Hjalmar MELLIN 1854--1933 2 I. INTRODUCTION 3 4 WHY? 5 6 . 7 II. BASICS 8 9 10 11 12 - - 13 III. HARMONIC SUMS Asymptotics 14


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Mellin vu du ciel Mellin, seen from the sky

Philippe Flajolet INRIA Rocquencourt

March 10, 2008

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Hjalmar MELLIN 1854--1933

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  • I. INTRODUCTION

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WHY?

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.

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  • II. BASICS

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  • -

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  • III. HARMONIC SUMS

Asymptotics

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<0,+oo> <1,+oo>

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<-1,0>

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<-2,-1>

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Digital trees (tries)

  • Set up recurrence and generating function
  • Solve and get a sum
  • Approximate (equiv. Poisson approx.)
  • Analyse harmonic sum

Digital trees aka “tries”

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  • IV. SOME GOODIES

A near(?) identity Möbius

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A (near) identity ???

  • Cf. Brigitte Vallée: binary Euclidean GCD

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A (near) identity ???

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1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, -1, . . .

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eorem: The sum tends to INFINITY like

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dist=10^(-5)

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[Clement+Flajolet-Vallee, 2000-2001]

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  • V. Advanced techniques

Perron’s formulae Approximating Dirichlet series Poisson + Mellin = Newton + Nörlund (Rice)

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  • VI. More goodies

Magic Duality and the Golden T riangle

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