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

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WHY? 5

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II. BASICS 8

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III. HARMONIC SUMS Asymptotics 14

<0,+oo> <1,+oo> 15

<-1,0> 16

<-2,-1> 17

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Digital trees (tries) Digital trees aka “tries” • Set up recurrence and generating function • Solve and get a sum • Approximate (equiv. Poisson approx.) • Analyse harmonic sum 19

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IV. SOME GOODIES A near(?) identity Möbius 23

A (near) identity ??? Cf. Brigitte Vallée: binary Euclidean GCD 24

A (near) identity ??? 25

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1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, -1, . . . 27

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� eorem : The sum tends to INFINITY like 29

dist=10^(-5) 30

[Clement+Flajolet-Vallee, 2000-2001] 31

V. Advanced techniques Perron’s formulae Approximating Dirichlet series Poisson + Mellin = Newton + Nörlund (Rice) 32

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VI. More goodies Magic Duality and the Golden T riangle 36

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