Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and...

37
Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie der reziproken Funktionen und Integrale ist ein centrales Gebiet, welches manche anderen Gebiete der Analysis miteinander verbindet. Hjalmar Mellin Reporter : Ilya Posov Saint-Petersburg State University Faculty of Mathematics and Mechanics Joint Advanced Student School 2004 1

Transcript of Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and...

Page 1: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Mellin transforms and asymptotics: Harmonic sums

Phillipe Flajolet,Xavier Gourdon,Philippe Dumas

Die Theorie der reziprokenFunktionen und Integrale ist

ein centrales Gebiet, welchesmanche anderen Gebiete der

Analysis miteinander verbindet.

Hjalmar MellinReporter :Ilya PosovSaint-Petersburg State UniversityFaculty of Mathematics and Mechanics

Joint Advanced Student School 20041

Page 2: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Contents

• Mellin transform and its basic properties• Direct mapping• Converse mapping• Harmonic sums

Joint Advanced Student School 20042

Page 3: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Robert Hjalmar Mellin

Born: 1854 in Liminka, Northern Ostrobothnia, FinlandDied: 1933

Joint Advanced Student School 20043

Page 4: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Some terminology

• Analytic function

• Meromorphic function

• Open strip

20 1 0 2 0( ) ( ) ( )f s c c s s c s s= + − + − +L

,a b⟨ ⟩

Joint Advanced Student School 20044

Page 5: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Mellin transform* 1

0

[ ( ); ] ( ) ( ) sM f x s f s f x x dx∞

−= = ∫u( ) O( ) 0f x x x= →v( ) O( )f x x x= → +∞

*( ) exists in the strip ,f s u v⟨− − ⟩

fundamental strip

Joint Advanced Student School 20045

Page 6: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Mellin transform example

1

1*

0

1( )1

1 O(1) 01

1 O( )1

( )1 sin

s

f xx

xx

x xx

xf s dxx s

ππ

+∞ −

=+

= →+

= → +∞+

= =+∫

fundamental strip :0 1, 0,1

u vu v= = −

⟨− − ⟩ = ⟨ ⟩

Joint Advanced Student School 20046

Page 7: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Gamma function

* 1

0

( )O(1) 0O( ) 0

( ) ( ) ( ) ( 1)

x

x

x b

x s

f x ee xe x b x

f s e x dx s s s s

− −

+∞− −

=

= →

= ∀ > → +∞

= = Γ Γ = Γ +∫

0, 0,

u vu v= = −∞

⟨− − ⟩ = ⟨ +∞⟩

fundamental strip :

Joint Advanced Student School 20047

Page 8: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Transform of step function

[ ]( )

1* 1 1

0 0

*

0, 0,1H( )

1, 1,

H( ) O(1) 0H( ) O( ) 0

1H ( ) ( ) , 0,

1H( ) 1 H( ), H ( ) , ,0

b

s s

xx

x

x xx x b x

x H x x dx x dx ss

x x x ss

+∞− −

⎧ ∈⎪= ⎨ ∈ +∞⎪⎩= →

= ∀ > → +∞

= = = ∈⟨ +∞⟩

= − = − ∈⟨−∞ ⟩

∫ ∫

Joint Advanced Student School 20048

Page 9: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Basic properties (1/2)

( )

*

*

*

*

*

*

( ) ( ) ,( ) ( ) ,

1( ) , 0

(1/ ) ( ) ,1( ) ( ) , 0

( ) ( )

s

sk k kk k

f x f sx f x f s

sf x f

f x f s

f x f s

f x f s

ν

ρ

α βν α ν β ν

ρα ρβ ρρ ρ

β α

µ α β µµ

λ µ λ µ−

⟨ ⟩+ ⟨ − − ⟩

⎛ ⎞⟨ ⟩ >⎜ ⎟

⎝ ⎠− − ⟨− − ⟩

⟨ ⟩ >

∑ ∑

Joint Advanced Student School 20049

Page 10: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Basic properties (2/2)*

*

*

*

*

0

( ) ( ) ,

( ) log ( ) ,

( ) ( ) ,

( ) ( 1) ( 1) 1, 1

1( ) ( 1)x

f x f sdf x x f sds

df x sf s xdx

d f x s f sdx

f t dt f ss

α β

α β

α β

α β

⟨ ⟩

⟨ ⟩

′ ′Θ − ⟨ ⟩ Θ =

′ ′− − − ⟨ + + ⟩

− +∫

Joint Advanced Student School 200410

Page 11: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Zeta function

( ) *

2 3

*

( ); ( )

( ) ...1

1, , ( )1 1 1( ) ; ( ) ( )1 2 3

1,

sk k kk k

xx x x

x

xk k

xs s s

M f x s f s

eg x e e ee

k f x e

g s M e s s s

s

λ µ λ µ

λ µ

ζ

−− − −

⎡ ⎤ =⎣ ⎦

= = + + +−

= = =

⎛ ⎞ ⎡ ⎤= + + + = Γ⎜ ⎟ ⎣ ⎦⎝ ⎠∈⟨ +∞⟩

∑ ∑

L

Joint Advanced Student School 200411

Page 12: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Some Mellin tranforms

( )2 1 12 2

0 1

1

( ) 0,1 ( ) 1,0

0,

( ) ( ) 1,1

1 0,11 sin

log(1 ) 1,0sin

1H( ) 1 0,

( 1) !(log ) H( ) ,( )

x

x

x

x

x

x

kk

k

e se s

e se s s

e

x s

xs s

xs

kx x x ks

α

ζ

ππππ

αα

< <

+

Γ ⟨ +∞⟩− Γ ⟨− ⟩

Γ ⟨ +∞⟩

Γ ⟨ +∞⟩−

⟨ ⟩+

+ ⟨− ⟩

≡ ⟨ +∞⟩

−⟨− +∞⟩ ∈

+Joint Advanced Student School 200412

Page 13: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Inversion

Theorem 1

*

i*

i

Let ( ) be integrable with fundamential strip , .Let and ( i ) is integrable, then the equality

1 ( ) ( )2 iholds almost everywhere.

cs

c

f xc f c t

f s x ds f x

α β

α β

π

+ ∞−

− ∞

⟨ ⟩

< < +

=∫

Joint Advanced Student School 200413

Page 14: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Laurent expansion

0

0

02

02 2

( ) ( )

0Pole of order if r>0Analytic in if 0Example :

1 1 2 3( 1) ( 1)( 1) 11 1 1 1 ( 0)

( 1)

kk

k r

r

s c s s

cr

s r

s ss s s

ss s s s

φ+∞

≥−

= −

= + + + + = −+ +

= − + + =+

L

L

Joint Advanced Student School 200414

Page 15: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Singular elementDefinition 1

0

2 20 1

0 2 2

A singular element of ( ) at is an initial sum ofLaurent expansion truncated at terms of O(1) or smaller :

1 1 1 1( ) 1 2 3( 1)( 1) 1

1 1 1 1s.e. at =0 : , 1

s s

s s

s ss s s s s

ss s s s

φ

φ→ →−

= − + − + + + ++ +

⎡ ⎤ ⎡ ⎤− − +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

= =L L

0

,

1 1 1s.e. at = 1 : , 2 , 2 3( 1) ,1 1 1

s ss s s⎡ ⎤ ⎡ ⎤ ⎡ ⎤− + + + +⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ + +⎣ ⎦ ⎣ ⎦ ⎣ ⎦

K

K

Joint Advanced Student School 200415

Page 16: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Singular expansionDefinition 2

2

Let ( ) be meromorfic in with including all the poles of ( ) in . A singular expansion of ( )in is a formal sum of singular elements of ( ) atall points of Notation : ( ) ( )

1 1( 1)

ss s

s

s E s

s s

φφ φ

φ

φ

Ω ℘Ω

Ω℘

∈Ω

+

^

^ 21 0 1

1 1 1 2,21 2s s s

ss s s=− = =

⎡ ⎤ ⎡ ⎤ ⎡ ⎤+ − + ∈⟨− ⟩⎢ ⎥ ⎢ ⎥ ⎢ ⎥+⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Joint Advanced Student School 200416

Page 17: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Singular expansion of gamma function

0

( ) ( 1)( 1)( )

( 1)( 2) ( )Thus ( ) has poles at the points with

( 1) 1( )!( 1) 1( )

!

m

k

k

s s ss ms

s s s s ms s m m

sm s m

s sk s k

+∞

=

Γ = Γ +Γ + +

Γ =+ + +

Γ = − ∈

−Γ

+−

Γ ∈+∑

L

^

poles of gamma function

Joint Advanced Student School 200417

Page 18: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Direct mapping (1/3)Theorem 2

*

( )

*

Let ( ) have a transform ( ) with nonempty fundamentalstrip .

( ) (log ) O( ) 0

,Then ( ) is continuable to a meromorphic function in thestrip where it admi

kk

k A

f x f s

f x c x x x x

kf s

ξ γξ

ξ

α β

γ ξ α

γ β

+,

, ∈

⟨ , ⟩

= + →

∈ − < − ≤

⟨− , ⟩

*1

( )

ts the singular expansion( 1) !( ) ( )

( )

k

k kk A

kf s c ssξ

ξ

γ βξ, +

, ∈

−∈⟨− , ⟩

+∑^

Joint Advanced Student School 200418

Page 19: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Direct mapping (2/3)*( ) ( )

Order at 0: O( ) Leftmost boundary of f.s. at ( )Order at + : O( ) Rightmost boundary of f.s. at ( )Expansion till O( ) at 0 Meromorphic continuation till ( )Expansion till O( )

f x f sx s

x sx sx

α

β

γ

δ

αβγ

ℜ =∞ ℜ =

ℜ ≥ −at + Meromorphic continuation till ( )s δ∞ ℜ ≤ −

Joint Advanced Student School 200419

Page 20: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Direct mapping (3/3)*

1

1

2

( ) ( )( 1) !Term ( log ) at 0 Pole with s.e.

( )( 1) !Term ( log ) at + Pole with s.e.

( )1Term at 0 Pole with s.e.

1Term log at 0 Pole with s.e. ( )

ka k

k

ka k

k

a

a

f x f skx x

s akx x

s a

xs a

x xs a

+

+

−+

−∞ −

+

+

−+

Joint Advanced Student School 200420

Page 21: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Example 0

( )2 3

1

0 0

*

*

M*

0

0

( 1)( ) 1 O2! 3! !

( ) ( ), 0,( ) is meromorphically continuable to 1,

( 1) 1( ) 1,!

finally( 1) 1( )

!

jMx j M

x j

j

j

j

j

x xf x e x x xj

f s s sf s M

f s s Mj s j

s sj s j

− +

→ =

=

+∞

=

−= = − + − + = +

= Γ ∈⟨ +∞⟩

⟨− − +∞⟩

−∈⟨− − +∞⟩

+

−Γ ∈

+

L

^

^

poles of gamma functionJoint Advanced Student School 200421

Page 22: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Example 1*

1 0 10 1

1

1

0

1

( ) , ( ) ( ) ( ) 1,1

1, 1, ,1 ( 1)! 2

1( ) ( )( 1)!

( 1) 1( )!

1 1( ) , , )1 2 1

result: ) is meromorph

x

x

x j

jxx j

j

j

j

j

m

ef x f s s s se

e xB B Be j

Bs s s

j s j

s sj s j

Bs s ms m

s

ζ

ζ

ζ ζ ζ

ζ

− +∞

+−→ =−

+∞+

=−

+∞

=

+

= = Γ ∈⟨ +∞⟩−

= = = −− +

Γ ∈+ +

−Γ ∈

+

∈ (0) = − (− = −− +(

L

^

^

^

0ic in with the only pole at 1s =

Joint Advanced Student School 200422

Page 23: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

0 0

1

1

*

*

0

1*

1

1( ) ( 1)1

1/ ( 1)1 1/

( ) , 0,1sin

( 1)( ) , ,1 (continuation to the left of the f.s.)

( 1)( ) , 0, (continuation to the ri

n n

x n

n n

x n

n

n

n

n

f x xx

x xx

f s sx

f s ss n

f s ss n

ππ

+

+∞

→ =

+∞− −

→+∞ =

+∞

=

−+∞

=

= = −+

= −+

= ∈⟨ ⟩

−∈⟨−∞ ⟩

+

−− ∈⟨ +∞⟩

^

^

*

ght of the f.s.)

( 1)( ) ( )sin

n

nf s s

x s nππ ∈

−≡ ∈

+∑^

Example 2

Joint Advanced Student School 200423

Page 24: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Example 3 (nonexplicit transform)

2 4 6 8 10

*

*

*

1 1 7 139 5473( ) 1 O( ) 04 96 5760 645120cosh

( ) has a fundamental strip 0,1 1 1 7 1 139 1 5473 1( ) 10,

4 2 96 4 5760 6 645120 81 1 1 7 1 139 1 5473 1( )

4 2 96 4 5760 6 645120

f x x x x x x xx

f s

f s ss s s s s

f ss s s s

= = − + − + + →

⟨ +∞⟩

− + − + ∈⟨− +∞⟩+ + + +

− + − ++ + +

^

^8

ss

+ ∈+

L

Joint Advanced Student School 200424

Page 25: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Converse mapping (1/3)Theorem 3

*

*

*

Let ( ) C(0, ) have a transform ( ) with nonempty fundamentalstrip .Let ( ) be meromorphically continuable to with a finitenumber of poles there, and be analytic on ( ) .

Let (

f x f s

f ss

f s

α β

γ βγ

∈ +∞⟨ , ⟩

⟨− , ⟩ℜ = −

( )) O with 1 when in .rs r s γ β−= > →+∞ ⟨− , ⟩

L

Joint Advanced Student School 200425

Page 26: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Converse mapping (2/3)Theorem 3 (continue)

*

( )

11

( )

1If ( )( )

,Then an asymptotic expansion of ( ) at 0 is

( 1)( ) (log ) O( )( 1)!

k kk A

kk

kk A

f x d ss

kf x

f x d x x xk

ξξ

ξ γξ

ξ

γ βξ

γ ξ β

,, ∈

−− −

,, ∈

∈⟨− , ⟩−

∈ − < − ≤

⎛ ⎞−= +⎜ ⎟−⎝ ⎠

L

^

Joint Advanced Student School 200426

Page 27: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Converse mapping (3/3)*( ) ( )

Pole at Term in asymptotic expansion left of f.s. expansion at 0 right of f.s expansion at +Simple pole

1 left: at 0

1 right: at +

Multiple pole Logarifmi

f s f xx

xs

xs

ξ

ξ

ξ

ξ

ξ

ξ

− ∞−

1

1

c factor1 ( 1) left: (log ) at 0

( ) !1 ( 1) right: (log ) at +

( ) !

kk

k

kk

k

x xs k

x xs k

ξ

ξ

ξ

ξ

−+

−+

−−

−− ∞

Joint Advanced Student School 200427

Page 28: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Example 4

0

/ 2 i

/ 2 i

1 2

0

( ) )) , analytic in strip 0,

( 1) ( 1) 1) ,! ( )

1( ) ( ) - original function2 i

( 1) ( 1)( ) O( ) 0! ( )

( ) (1 ) has the

j

j

s

jMj M

j

s ss

js sj s j

f x s x ds

jf x x x xj

f x x

ν

ν

ν

νφ νν

νφ νν

φπ

νν

+∞

=

+ ∞−

− ∞

+

=

Γ Γ( −( = ⟨ ⟩

Γ( )

− Γ + −( ∈⟨−∞ ⟩

Γ +

=

− Γ + −= + →

Γ

= +

^

same expansion at 0( ) (1 ) ( ), where ( ) O( ) 0Mf x x x x x Mν ϖ ϖ−= + + = ∀ >

Joint Advanced Student School 200428

Page 29: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Example 5

0

0

1 2

0

0

( ) (1 ) 0,1sin

1( ) ( 1) ! ,1

( ) ( 1) ! - the expansion is only asymptotic!

( ) ( 1) ! O( ) 0

In fact ( )1

n

n

n n

n

Mn n M

n

t

s s ss

s n ss n

f x n x

f x n x x x

ef x dtxt

πφπ

φ+∞

=

+∞

=

+

=

+∞ −

= Γ − ∈⟨ ⟩

− ∈⟨−∞ ⟩+

⎛ ⎞= − + →⎜ ⎟⎝ ⎠⎛ ⎞

=⎜ ⎟+⎝ ⎠

^

Joint Advanced Student School 200429

Page 30: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Harmonic sumsDefinition 3

[ ] *

* *

( ) ( ) harmonic sum

amplitudes

frequencies

) The Dirichlet series

( ); ( )

( ) ( ) ( )

j kj

j

j

sj j

j

s

G x g x

s

M g x s g s

G s s g s

λ µ

λ

µ

λ µ

µ µ

=

Λ( =

=

= Λ

∑(separation property)

Joint Advanced Student School 200430

Page 31: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Harmonic sum formula

k

The Mellin transform of the harmonic sum( ) ( ) is defined in the intersection

of the the fundamental strip of g( ) and thedomain of absolute convergence of the Dirichletseries ( ) (w

k kk

sk k

G x g x

x

s

λ µ

λ µ −

=

Λ =

* *

hich is of the form

( ) for some real )

( ) ) ( )a as

G s s g s

σ σℜ >

= Λ(

Joint Advanced Student School 200431

Page 32: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Example 61 1

1

1 1

* *

*2

1 1 1 / 1 1( ) , , , ( )1 / 1

1 1 1( ) 12 3

( ) (1 )

( ) ( ) ( ) ) 1,0sin

1 1( ) , (1 )1

1( )

k kk k

n

s sk k

k k

x k xh x g xk k x k x k k k x

h n Hn

s k s

h s s g s s ss

s ss s

h ss s

λ µ

λ µ ζ

πζπ

ζ γ ζ γ

γ

+∞ +∞

= =

+∞ +∞− − +

= =

⎡ ⎤= − = = = =⎢ ⎥+ + +⎣ ⎦

= = + + + +

Λ = = = −

= Λ = − (1− ∈⟨− ⟩

= + + − = − +−⎡ ⎤−⎢ ⎥⎣ ⎦

∑ ∑

∑ ∑

L

L

^1

41

1 )( 1) 1,

( 1) 1 1 1 1log log2 12 120

k

k

kk

n kk

k ss k

BH n nk n n n n

ζ

γ γ

+∞

=

( −− − ∈⟨− +∞⟩

−+ + + + − + +

∑ L

Joint Advanced Student School 200432

Page 33: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Example 7

1

1 1

* *

*2 2

( ) log 1) log 1 2,

11, , ( ) log(1 )

) )

( ) ) ( ) ) , 2, 1sin

1 1 1 log 2 (( )( 1) 1) 2

n

n n

s sk k

n n

x xl x x x sn n

g x x xn

s n s

l s s g s s ss s

l ss s s s

γ

λ µ

λ µ ζ

πζπ

γ π

+∞

=

+∞ +∞−

= =

⎡ ⎤⎛ ⎞= Γ( + − = − + ∈< − +∞ >⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

= = = − +

Λ( = = = (−

= Λ( = − (− ∈⟨− − ⟩

⎡ ⎤⎡ ⎤−+ + − +⎢ ⎥⎢ ⎥+ ( +⎣ ⎦ ⎣ ⎦

∑ ∑

^

( )

1

1

22 1

1

22 1

1

1) )( )

1 1( ) log( !) log (1 log log 22 2 (2 1)

1log( !) log 22 (2 1)

n

n

nn

n

x x nn

n

nn s n

Bl x x x x x x xn n x

Bx x e xn n x

ζ

γ γ π

π

−+∞

=

+∞

−=

+∞−

−=

− (−+

= − = − − ) − + +−

= +−

Simple poles in positive integersDouble poles in 0 and -1

Joint Advanced Student School 200433

Page 34: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Example 81

1

*

1* 1

20

1

( ) (polylogarithm ( ) )

1 , , ( )

) ( ), ( ) ( ) ) 1,

) 1 ( 1) 1( ) ( 1)! ( 1)! ( 1) 1

nxn w

w ww nn

xn kw

k

kn k

kn

n k

eL x Li z z n w kn

n g x en

s s k L s s k s s

k n HL sn s n k s k s k

λ µ

ζ ζ

ζ

−+∞+∞ −=

=

−+∞−

=≠ −

= = = ∈

= = =

Λ( = + = + Γ( ∈⟨ +∞⟩

⎡ ⎤( − −− + +⎢ ⎥+ − + − + −⎣ ⎦

∑ ∑

∑^

11

10

1

12

1 20

,( 1) )( ) [ log ] ( 1)( 1)! !

( )( ) ( 1)

!

kk n n

k kn

n k

n n

n

sk nL x x x H x

k n

nL x x

x n

ζ

ζπ

− +∞−

−=≠ −

+∞

=

∈⟨−∞ +∞⟩

− ( −= − + + −

−= + −

∑Joint Advanced Student School 200434

Page 35: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Example 9modified theta function

2 22

1

*1

*1 2

2 41

*0

1( ) , , , ( )

1( ) 1) 0,2 2

double pole at 0 and simple poles at 21 1 1 1 1( )

12 2 240 41 1( ) log 0

12 2401( )

n xx

w n nw wn

ex n g x en nss s s

s s m

s ss s s s

x x x x x

s

λ µ

ζ

γ

γ

−+∞−

=

Θ = = = =

⎛ ⎞Θ = Γ ( + ∈⟨ +∞⟩⎜ ⎟⎝ ⎠

= = −

⎡ ⎤Θ + + + + ∈⎢ ⎥2 + +⎣ ⎦

Θ = − + + + + →2

Θ =

L

L

^

01 1) O( ) 0

2 2 2 2Ms s x x

xπζ⎛ ⎞Γ ( Θ = − + →⎜ ⎟

⎝ ⎠Joint Advanced Student School 200435

Page 36: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

Example 10

( )

( )

1

1 1

*

2*

20

22 1

0

( ) ( ) , ( ), , ( )

( )) ( )

( ) ) ( )

1 1 ( 2 1) 1( )( 1) 1 4 2 1 ! 2 1

1 1 ( 2 1)( ) ( log4 2 1 !

kx xk k

k

sk k s

k k

k

k

k

D x d k e d k k g x e

d ks sk

D s s s

kD ss s s k s k

kD x x xx k

λ µ

λ µ ζ

ζ

γ ζ

ζγ

+∞− −

=

+∞ +∞− 2

−= =

2

+∞

=

+∞+

=

= = = =

Λ( = = =

= Γ(

⎡ ⎤ − −⎡ ⎤+ + −⎢ ⎥ ⎢ ⎥− − + + +⎣ ⎦⎣ ⎦− −

− + ) + −+

∑ ∑

^

Joint Advanced Student School 200436

Page 37: Mellin Transform and asymptotics: Harmonic sums · 2004. 11. 24. · Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie

The end

Joint Advanced Student School 200437