Poisson-Newton formulas and Dirichlet series - matematicas… · Classical Poisson formulaDirichlet seriesPoisson formulas for Dirichlet seriesProof of TheoremFurther results Poisson-Newton

Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results

Poisson-Newton formulas and Dirichlet series

Vicente Muñoz (UCM)

27 de noviembre de 2012

Universidad Carlos III de Madrid

V. Muñoz UCM



Poisson formulas and Dirichlet series

1 Classical Poisson formula

2 Dirichlet series

3 Poisson formulas for Dirichlet series

4 Proof of Theorem

5 Further results

(Joint work with Ricardo Pérez-Marco.)

V. Muñoz UCM



Classical Poisson formula

The Poisson formula reads∑n∈Z

e2πint =∑k∈Z

δk

The waves with frequences λn = n,n = 0,1,2, . . . are resonant at theintegers k ∈ Z.

The fundalmental frequency isλ1 = 1.

V. Muñoz UCM





e2πint =∑k∈Z

δk



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e2πint =∑k∈Z

δk



V. Muñoz UCM



Dirac delta

The Dirac delta is defined as:

δa =

{0, x 6= a∞, x = a

where ∫Rδa = 1

This is not a function.

So what is this?

V. Muñoz UCM



Dirac delta


δa =

{0, x 6= a∞, x = a

where ∫Rδa = 1


So what is this?

V. Muñoz UCM



Dirac delta


δa =

{0, x 6= a∞, x = a

where ∫Rδa = 1


So what is this?

V. Muñoz UCM



Distributions

We interpret a function as a functional:

〈f ,g〉 =

∫ ∞−∞

f (x)g(x)dx

So the Dirac delta is:

〈δa,g〉 = g(a)

The distributions are (continuous) functionals on our space offunctions.We use exponentially decaying test functions: |g(x)| ≤ Ce−κ|x |

V. Muñoz UCM



Distributions


〈f ,g〉 =

∫ ∞−∞

f (x)g(x)dx


〈δa,g〉 = g(a)


V. Muñoz UCM



Distributions


〈f ,g〉 =

∫ ∞−∞

f (x)g(x)dx


〈δa,g〉 = g(a)

The distributions are (continuous) functionals on our space offunctions.

We use exponentially decaying test functions: |g(x)| ≤ Ce−κ|x |

V. Muñoz UCM



Distributions


〈f ,g〉 =

∫ ∞−∞

f (x)g(x)dx


〈δa,g〉 = g(a)


V. Muñoz UCM



Fourier transform

The Fourier transform

f (x) = e2πiαx f (t) = δα

A “basis” of the “space” of functions is given by{e2πiαx ; α ∈ R}, and another one is {δα ; α ∈ R}So

f (x) =∑α∈R

aαe2πiαx ∑α∈R

aαδα

Rewriting, we get the inverse Fourier transform

a(x) =

∫α∈R

a(α)e2πiαxdα ! a(α)

V. Muñoz UCM



Fourier transform



A “basis” of the “space” of functions is given by{e2πiαx ; α ∈ R},

and another one is {δα ; α ∈ R}So

f (x) =∑α∈R


aαδα


a(x) =

∫α∈R


V. Muñoz UCM



Fourier transform



A “basis” of the “space” of functions is given by{e2πiαx ; α ∈ R}, and another one is {δα ; α ∈ R}

Sof (x) =

∑α∈R


aαδα


a(x) =

∫α∈R


V. Muñoz UCM



Fourier transform




f (x) =∑α∈R


aαδα


a(x) =

∫α∈R


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Fourier transform




f (x) =∑α∈R


aαδα


a(x) =

∫α∈R


V. Muñoz UCM



Poisson formula

The Poisson formula∑n∈Z

e2πint =∑k∈Z

δk is now rewritten as

∑n∈Z

∫ ∞−∞

e 2πinxg(x)dx =∑k∈Z

g(k)

for any test function g.

The alternative form∑n∈Z

g(n) =∑k∈Z

g(k)

is the original form of the Poissonformula.

Poisson (1781-1840)

V. Muñoz UCM



Poisson formula

The Poisson formula∑n∈Z

e2πint =∑k∈Z

δk is now rewritten as

∑n∈Z

∫ ∞−∞

e 2πinxg(x)dx =∑k∈Z

g(k)

for any test function g.

The alternative form∑n∈Z

g(n) =∑k∈Z

g(k)

is the original form of the Poissonformula.

Poisson (1781-1840)V. Muñoz UCM



Reinterpreting the classical Poisson formula

Let λ > 0 be the fundamental frequency.

Then the Poisson formula reads∑n∈Z

e2πiλ

nt = λ∑k∈Z

δλk

This is associated to the Dirichlet series f (s) = 1− e−λs

Left-hand-side:∑

eρt , over zeroes ρ = −2πiλ n of f .

Right-hand-side: Dirac deltas at multiples of fundamentalfrequency of f .

V. Muñoz UCM




Let λ > 0 be the fundamental frequency.Then the Poisson formula reads∑

n∈Ze

2πiλ

nt = λ∑k∈Z

δλk


Left-hand-side:∑



V. Muñoz UCM





n∈Ze

2πiλ

nt = λ∑k∈Z

δλk


Left-hand-side:∑



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n∈Ze

2πiλ

nt = λ∑k∈Z

δλk


Left-hand-side:∑



V. Muñoz UCM





n∈Ze

2πiλ

nt = λ∑k∈Z

δλk


Left-hand-side:∑



V. Muñoz UCM



Dirichlet series

Definition

A Dirichlet series is f (s) = 1 +∑

ane−λns, 0 < λ1 < λ2 < . . .,an ∈ C, and there exists some σ ∈ R such that∑|an|e−λnσ <∞.

Dirichlet (1805-1859)

We shall assume that the Dirichletseries has a meromorphic extensionto the whole of C.

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Dirichlet series

Definition





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Dirichlet series

Definition





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Riemann zeta function


ζ(s) =∞∑

n=1

1ns = 1 +

∑n≥2

e−(log n)s

Riemann (1826-1866)

Riemann (1859) usedζ(s) to study the num-ber of primes π(x) upto x ∈ R.

V. Muñoz UCM




The Riemann zeta function hasA single pole at s = 1.

Zeroes at s = −2,−4,−6, . . . (trivial zeroes).Other zeroes with 0 < <s < 1 (the critical strip).

ξ(s) = 12π−s/2s(s − 1)Γ(s/2)ζ(s) satisfies

the functional equation

ξ(1− s) = ξ(s).

So the zeroes are symmetric with respectto <s = 1/2.

V. Muñoz UCM




The Riemann zeta function hasA single pole at s = 1.Zeroes at s = −2,−4,−6, . . . (trivial zeroes).

Other zeroes with 0 < <s < 1 (the critical strip).



ξ(1− s) = ξ(s).


V. Muñoz UCM




The Riemann zeta function hasA single pole at s = 1.Zeroes at s = −2,−4,−6, . . . (trivial zeroes).Other zeroes with 0 < <s < 1 (the critical strip).



ξ(1− s) = ξ(s).


V. Muñoz UCM







ξ(1− s) = ξ(s).


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ξ(1− s) = ξ(s).


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ξ(1− s) = ξ(s).


V. Muñoz UCM



Riemann hypothesis

Riemann hypothesis

All non-trivial zeroes of ζ satisfy <s = 1/2.

First zeroes: ρk = 1/2 + iγk , where γk =14,13472521,02204025,01085830,42487632,935062, . . .

One of the most important open problems in mathematics!The oldest one in the list of the Millenium problems.

V. Muñoz UCM



Riemann hypothesis

Riemann hypothesis




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Riemann hypothesis

Riemann hypothesis




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Riemann hypothesis

Riemann hypothesis




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Poisson formula for a Dirichlet series

Let f (s) = 1 +∑

ane−λns.

Write

− log f (s) =∑

b~r e−(λ1r1+...+λ`r`)s =

∑b~r e−〈~λ,~r 〉s

Theorem (V. M. & R. Pérez-Marco)

Let D =∑

nρρ be the divisor of zeroes/poles of f . Then, asdistributions, ∑

nρeρt =∑〈~λ,~r 〉b~r δ〈~λ,~r 〉

on R+

W (t) =∑

nρeρt is called the Newton-Cramer distribution.

V. Muñoz UCM




Let f (s) = 1 +∑

ane−λns. Write

− log f (s) =∑

b~r e−(λ1r1+...+λ`r`)s

=∑

b~r e−〈~λ,~r 〉s


Let D =∑



on R+

W (t) =∑


V. Muñoz UCM




Let f (s) = 1 +∑

ane−λns. Write

− log f (s) =∑

b~r e−(λ1r1+...+λ`r`)s =

∑b~r e−〈~λ,~r 〉s


Let D =∑



on R+

W (t) =∑


V. Muñoz UCM




Let f (s) = 1 +∑

ane−λns. Write

− log f (s) =∑

b~r e−(λ1r1+...+λ`r`)s =

∑b~r e−〈~λ,~r 〉s


Let D =∑



on R+

W (t) =∑


V. Muñoz UCM




Let f (s) = 1 +∑

ane−λns. Write

− log f (s) =∑

b~r e−(λ1r1+...+λ`r`)s =

∑b~r e−〈~λ,~r 〉s


Let D =∑



on R+

W (t) =∑


V. Muñoz UCM




Let f (s) = 1 +∑

ane−λns. Write

− log f (s) =∑

b~r e−(λ1r1+...+λ`r`)s =

∑b~r e−〈~λ,~r 〉s


Let D =∑



on R+

W (t) =∑


V. Muñoz UCM



Example: the zeta function

ζ(s) =∑n≥1

1ns

=∏

p

(1 +

1ps +

1p2s + . . .

)=∏

p

(1− 1

ps

)−1

− log ζ(s) =∑

p

log(1− p−s)

=∑

p

∑k≥1

−1k

p−ks

=∑p,k

−1k

e−(k log p)s

So∑nρeρt = −

∑p,k

(log p)δk log p

V. Muñoz UCM




ζ(s) =∑n≥1

1ns =

∏p

(1 +

1ps +

1p2s + . . .

)

=∏

p

(1− 1

ps

)−1

− log ζ(s) =∑

p

log(1− p−s)

=∑

p

∑k≥1

−1k

p−ks

=∑p,k

−1k

e−(k log p)s

So∑nρeρt = −

∑p,k

(log p)δk log p

V. Muñoz UCM




ζ(s) =∑n≥1

1ns =

∏p

(1 +

1ps +

1p2s + . . .

)=∏

p

(1− 1

ps

)−1

− log ζ(s) =∑

p

log(1− p−s)

=∑

p

∑k≥1

−1k

p−ks

=∑p,k

−1k

e−(k log p)s

So∑nρeρt = −

∑p,k

(log p)δk log p

V. Muñoz UCM




ζ(s) =∑n≥1

1ns =

∏p

(1 +

1ps +

1p2s + . . .

)=∏

p

(1− 1

ps

)−1

− log ζ(s) =∑

p

log(1− p−s)

=∑

p

∑k≥1

−1k

p−ks

=∑p,k

−1k

e−(k log p)s

So∑nρeρt = −

∑p,k

(log p)δk log p

V. Muñoz UCM




ζ(s) =∑n≥1

1ns =

∏p

(1 +

1ps +

1p2s + . . .

)=∏

p

(1− 1

ps

)−1

− log ζ(s) =∑

p

log(1− p−s)

=∑

p

∑k≥1

−1k

p−ks

=∑p,k

−1k

e−(k log p)s

So∑nρeρt = −

∑p,k

(log p)δk log p

V. Muñoz UCM




ζ(s) =∑n≥1

1ns =

∏p

(1 +

1ps +

1p2s + . . .

)=∏

p

(1− 1

ps

)−1

− log ζ(s) =∑

p

log(1− p−s)

=∑

p

∑k≥1

−1k

p−ks

=∑p,k

−1k

e−(k log p)s

So∑nρeρt = −

∑p,k

(log p)δk log p

V. Muñoz UCM




ζ(s) =∑n≥1

1ns =

∏p

(1 +

1ps +

1p2s + . . .

)=∏

p

(1− 1

ps

)−1

− log ζ(s) =∑

p

log(1− p−s)

=∑

p

∑k≥1

−1k

p−ks

=∑p,k

−1k

e−(k log p)s

So∑nρeρt = −

∑p,k

(log p)δk log p

V. Muñoz UCM



Cramer function

Write ρ = 12 + iγ, for the non-trivial zeroes.

Let W0(t) = −et +∑n≥1

e−2nt = −et +e−2t

1− e−2t

et/2∑

eiγt + W0(t) = −∑p,k

(log p)δk log p

The Cramer function is

V (t) =∑γ>0

eiγt

Then

et/2V (t) + et/2V (−t) + W0(t) = −∑p,k

(log p)δk log p

V. Muñoz UCM



Cramer function


Let W0(t) = −et +∑n≥1


1− e−2t

et/2∑


(log p)δk log p


V (t) =∑γ>0

eiγt

Then

et/2V (t) + et/2V (−t) + W0(t) = −∑p,k

(log p)δk log p

V. Muñoz UCM



Cramer function


Let W0(t) = −et +∑n≥1


1− e−2t

et/2∑


(log p)δk log p


V (t) =∑γ>0

eiγt

Then

et/2V (t) + et/2V (−t) + W0(t) = −∑p,k

(log p)δk log p

V. Muñoz UCM



Cramer function


Let W0(t) = −et +∑n≥1


1− e−2t

et/2∑


(log p)δk log p


V (t) =∑γ>0

eiγt

Then

et/2V (t) + et/2V (−t) + W0(t) = −∑p,k

(log p)δk log p

V. Muñoz UCM



Cramer function


Let W0(t) = −et +∑n≥1


1− e−2t

et/2∑


(log p)δk log p


V (t) =∑γ>0

eiγt

Then

et/2V (t) + et/2V (−t) + W0(t) = −∑p,k

(log p)δk log p

V. Muñoz UCM



Newton formulas

Let P(z) = zn + a1zn−1 + . . .+ an be a polynomial, and letα1, . . . , αn be its zeroes.

Set z = es. We get a Dirichlet series

f (s) = e−nsP(es) = 1 + a1e−s + . . .+ ane−ns

A zero of f is of the form

eρjk = αje2πik

where j = 1, . . . ,n, k ∈ Z.

V. Muñoz UCM



Newton formulas

Let P(z) = zn + a1zn−1 + . . .+ an be a polynomial, and letα1, . . . , αn be its zeroes.Set z = es. We get a Dirichlet series



eρjk = αje2πik

where j = 1, . . . ,n, k ∈ Z.

V. Muñoz UCM



Newton formulas

Let P(z) = zn + a1zn−1 + . . .+ an be a polynomial, and letα1, . . . , αn be its zeroes.Set z = es. We get a Dirichlet series



eρjk = αje2πik

where j = 1, . . . ,n, k ∈ Z.

V. Muñoz UCM



Newton formulas

The Newton-Cramer distribution is

W (f ) =∑

eρt =∑j,k

αtj e

2πikt

=∑

j

αtj

∑k

e2πikt

=∑

j

αtj

∑m

δm (by the Poisson formula)

=∑

m

Smδm (where Sm =∑

j αmj are the Newton sums)

W (f )(t) generalizes the Newton sums to exponents t ∈ R+,and they are non-zero only for t = m ∈ Z+.

Poisson-Newton formula ⇐⇒ Newton relations for P(z).

V. Muñoz UCM



Newton formulas


W (f ) =∑

eρt =∑j,k

αtj e

2πikt

=∑

j

αtj

∑k

e2πikt

=∑

j

αtj

∑m


=∑

m





V. Muñoz UCM



Newton formulas


W (f ) =∑

eρt =∑j,k

αtj e

2πikt

=∑

j

αtj

∑k

e2πikt

=∑

j

αtj

∑m


=∑

m





V. Muñoz UCM



Newton formulas


W (f ) =∑

eρt =∑j,k

αtj e

2πikt

=∑

j

αtj

∑k

e2πikt

=∑

j

αtj

∑m


=∑

m





V. Muñoz UCM



Newton formulas


W (f ) =∑

eρt =∑j,k

αtj e

2πikt

=∑

j

αtj

∑k

e2πikt

=∑

j

αtj

∑m


=∑

m





V. Muñoz UCM



Newton formulas


W (f ) =∑

eρt =∑j,k

αtj e

2πikt

=∑

j

αtj

∑k

e2πikt

=∑

j

αtj

∑m


=∑

m





V. Muñoz UCM



Defining the Newton-Cramer distribution

The exponent of convergence is the minimum integer d ≥ 0such that

∑|nρ| |ρ|−d <∞.

Fix σ ∈ C and let

Kd (t) =∑ nρ

(ρ− σ)d (e(ρ−σ)t − 1)1R+

which is a continuous function. Define

W (f ) = eσt Dd

Dtd Kd (t)

W (f ) does not depend on σ over R+.There is a contribution at zero.

V. Muñoz UCM





∑|nρ| |ρ|−d <∞. Fix σ ∈ C and let

Kd (t) =∑ nρ

(ρ− σ)d (e(ρ−σ)t − 1)1R+

which is a continuous function.

Define

W (f ) = eσt Dd

Dtd Kd (t)


V. Muñoz UCM






Kd (t) =∑ nρ

(ρ− σ)d (e(ρ−σ)t − 1)1R+


W (f ) = eσt Dd

Dtd Kd (t)


V. Muñoz UCM






Kd (t) =∑ nρ

(ρ− σ)d (e(ρ−σ)t − 1)1R+


W (f ) = eσt Dd

Dtd Kd (t)

W (f ) does not depend on σ over R+.

There is a contribution at zero.

V. Muñoz UCM






Kd (t) =∑ nρ

(ρ− σ)d (e(ρ−σ)t − 1)1R+


W (f ) = eσt Dd

Dtd Kd (t)


V. Muñoz UCM



Hadamard interpolation

Let Em(z) = (1− z)ez+ 12 z2+... 1

m zmbe the Weierstrass factor.

Hadamard interpolation says that

f (s) = eQf (s)∏ρ

Ed−1

(s − σρ− σ

)nρ

The genus of f is g = m«ax{deg Qf ,d − 1}.Let

G(s) =∑ρ

nρ

(1

ρ− s−

d−2∑`=0

(s − σ)`

(ρ− σ)`+1

)Then

f ′

f= (log f )′ = Q′f + G

V. Muñoz UCM




Let Em(z) = (1− z)ez+ 12 z2+... 1




Ed−1

(s − σρ− σ

)nρ


G(s) =∑ρ

nρ

(1

ρ− s−

d−2∑`=0

(s − σ)`

(ρ− σ)`+1

)Then

f ′

f= (log f )′ = Q′f + G

V. Muñoz UCM




Let Em(z) = (1− z)ez+ 12 z2+... 1




Ed−1

(s − σρ− σ

)nρ

The genus of f is g = m«ax{deg Qf ,d − 1}.

Let

G(s) =∑ρ

nρ

(1

ρ− s−

d−2∑`=0

(s − σ)`

(ρ− σ)`+1

)Then

f ′

f= (log f )′ = Q′f + G

V. Muñoz UCM




Let Em(z) = (1− z)ez+ 12 z2+... 1




Ed−1

(s − σρ− σ

)nρ


G(s) =∑ρ

nρ

(1

ρ− s−

d−2∑`=0

(s − σ)`

(ρ− σ)`+1

)

Thenf ′

f= (log f )′ = Q′f + G

V. Muñoz UCM




Let Em(z) = (1− z)ez+ 12 z2+... 1




Ed−1

(s − σρ− σ

)nρ


G(s) =∑ρ

nρ

(1

ρ− s−

d−2∑`=0

(s − σ)`

(ρ− σ)`+1

)Then

f ′

f= (log f )′ = Q′f + G

V. Muñoz UCM



Laplace transform of W (f )

L(W (f )) = 〈W (f ),e−st〉

=

⟨Dd

Dtd Kd (t),e(σ−s)t⟩

=

∫ ∞0

(−1)d∑ρ

nρ1

(ρ− σ)d (e(ρ−σ)t − 1)dd

dtd e(σ−s)tdt

=∑ρ

nρ(s − σ)d

(ρ− σ)d

∫ ∞0

(e(ρ−s)t − e(σ−s)t )dt

= −∑ρ

nρ(s − σ)d

(ρ− σ)d

(1

ρ− s− 1σ − s

)= G(s)

V. Muñoz UCM




L(W (f )) = 〈W (f ),e−st〉

=

⟨Dd


=

∫ ∞0

(−1)d∑ρ

nρ1

(ρ− σ)d (e(ρ−σ)t − 1)dd

dtd e(σ−s)tdt

=∑ρ

nρ(s − σ)d

(ρ− σ)d

∫ ∞0


= −∑ρ

nρ(s − σ)d

(ρ− σ)d

(1

ρ− s− 1σ − s

)= G(s)

V. Muñoz UCM




L(W (f )) = 〈W (f ),e−st〉

=

⟨Dd


=

∫ ∞0

(−1)d∑ρ

nρ1

(ρ− σ)d (e(ρ−σ)t − 1)dd

dtd e(σ−s)tdt

=∑ρ

nρ(s − σ)d

(ρ− σ)d

∫ ∞0


= −∑ρ

nρ(s − σ)d

(ρ− σ)d

(1

ρ− s− 1σ − s

)= G(s)

V. Muñoz UCM




L(W (f )) = 〈W (f ),e−st〉

=

⟨Dd


=

∫ ∞0

(−1)d∑ρ

nρ1

(ρ− σ)d (e(ρ−σ)t − 1)dd

dtd e(σ−s)tdt

=∑ρ

nρ(s − σ)d

(ρ− σ)d

∫ ∞0


= −∑ρ

nρ(s − σ)d

(ρ− σ)d

(1

ρ− s− 1σ − s

)= G(s)

V. Muñoz UCM




L(W (f )) = 〈W (f ),e−st〉

=

⟨Dd


=

∫ ∞0

(−1)d∑ρ

nρ1

(ρ− σ)d (e(ρ−σ)t − 1)dd

dtd e(σ−s)tdt

=∑ρ

nρ(s − σ)d

(ρ− σ)d

∫ ∞0


= −∑ρ

nρ(s − σ)d

(ρ− σ)d

(1

ρ− s− 1σ − s

)

= G(s)

V. Muñoz UCM




L(W (f )) = 〈W (f ),e−st〉

=

⟨Dd


=

∫ ∞0

(−1)d∑ρ

nρ1

(ρ− σ)d (e(ρ−σ)t − 1)dd

dtd e(σ−s)tdt

=∑ρ

nρ(s − σ)d

(ρ− σ)d

∫ ∞0


= −∑ρ

nρ(s − σ)d

(ρ− σ)d

(1

ρ− s− 1σ − s

)= G(s)

V. Muñoz UCM



Proof of Poisson-Newton formula

Fromf ′

f= (log f )′ = Q′f + G

= (c0 + c1s + . . .+ cg−1sg−1) + G

take inverse Laplace tranform to get

W (f ) = L−1(G) = −g−1∑j=1

cjδj0 + L−1(f ′/f )

For a Dirichlet series f , log f =∑

b~r e−〈~λ,~r 〉s. Hence

L−1(log f ) =∑

b~rδ〈~λ,~r 〉and

L−1((log f )′) = s∑

b~rδ〈~λ,~r 〉 =∑

b~r 〈~λ,~r 〉 δ〈~λ,~r 〉 QED

V. Muñoz UCM




Fromf ′

f= (log f )′ = Q′f + G = (c0 + c1s + . . .+ cg−1sg−1) + G


W (f ) = L−1(G) = −g−1∑j=1

cjδj0 + L−1(f ′/f )



L−1(log f ) =∑


L−1((log f )′) = s∑

b~rδ〈~λ,~r 〉 =∑

b~r 〈~λ,~r 〉 δ〈~λ,~r 〉 QED

V. Muñoz UCM




Fromf ′



W (f ) = L−1(G)

= −g−1∑j=1

cjδj0 + L−1(f ′/f )



L−1(log f ) =∑


L−1((log f )′) = s∑

b~rδ〈~λ,~r 〉 =∑

b~r 〈~λ,~r 〉 δ〈~λ,~r 〉 QED

V. Muñoz UCM




Fromf ′



W (f ) = L−1(G) = −g−1∑j=1

cjδj0 + L−1(f ′/f )



L−1(log f ) =∑


L−1((log f )′) = s∑

b~rδ〈~λ,~r 〉 =∑

b~r 〈~λ,~r 〉 δ〈~λ,~r 〉 QED

V. Muñoz UCM




Fromf ′



W (f ) = L−1(G) = −g−1∑j=1

cjδj0 + L−1(f ′/f )


b~r e−〈~λ,~r 〉s.

Hence

L−1(log f ) =∑


L−1((log f )′) = s∑

b~rδ〈~λ,~r 〉 =∑

b~r 〈~λ,~r 〉 δ〈~λ,~r 〉 QED

V. Muñoz UCM




Fromf ′



W (f ) = L−1(G) = −g−1∑j=1

cjδj0 + L−1(f ′/f )



L−1(log f ) =∑

b~rδ〈~λ,~r 〉

and

L−1((log f )′) = s∑

b~rδ〈~λ,~r 〉 =∑

b~r 〈~λ,~r 〉 δ〈~λ,~r 〉 QED

V. Muñoz UCM




Fromf ′



W (f ) = L−1(G) = −g−1∑j=1

cjδj0 + L−1(f ′/f )



L−1(log f ) =∑


L−1((log f )′) = s∑

b~rδ〈~λ,~r 〉 =∑

b~r 〈~λ,~r 〉 δ〈~λ,~r 〉

QED

V. Muñoz UCM




Fromf ′



W (f ) = L−1(G) = −g−1∑j=1

cjδj0 + L−1(f ′/f )



L−1(log f ) =∑


L−1((log f )′) = s∑

b~rδ〈~λ,~r 〉 =∑

b~r 〈~λ,~r 〉 δ〈~λ,~r 〉 QED

V. Muñoz UCM



On the genus of Dirichlet series

Corollary (V. M. & R. Pérez-Marco)

For a Dirichlet series, we have

d = g + 1d ≥ 2

(Just compare the order of the two distributions in the two sidesof the Poisson-Newton formula)

V. Muñoz UCM





For a Dirichlet series, we haved = g + 1

d ≥ 2


V. Muñoz UCM





For a Dirichlet series, we haved = g + 1d ≥ 2


V. Muñoz UCM







V. Muñoz UCM







V. Muñoz UCM



Symmetric Poisson-Newton formula

For f (s) = 1 +∑

ane−λns, an ∈ R, we have∑nρeρ|t | = 2

∑c2lδ

(2l)0 +

∑~r∈Λ∪(−Λ)

b|~r |〈~λ, |~r |〉δ〈~λ,~r〉

on R.

For the classical Poisson formula:∑n∈Z

e2πiλ

nt = λδ0 +∑k∈Z∗

λδkλ

We have

c0 = λ/2bk = 1/k , k ∈ Z∗

V. Muñoz UCM




For f (s) = 1 +∑


∑c2lδ

(2l)0 +

∑~r∈Λ∪(−Λ)

b|~r |〈~λ, |~r |〉δ〈~λ,~r〉

on R.For the classical Poisson formula:∑

n∈Ze

2πiλ


λδkλ

We have

c0 = λ/2bk = 1/k , k ∈ Z∗

V. Muñoz UCM




For f (s) = 1 +∑


∑c2lδ

(2l)0 +

∑~r∈Λ∪(−Λ)

b|~r |〈~λ, |~r |〉δ〈~λ,~r〉


n∈Ze

2πiλ


λδkλ

We havec0 = λ/2

bk = 1/k , k ∈ Z∗

V. Muñoz UCM




For f (s) = 1 +∑


∑c2lδ

(2l)0 +

∑~r∈Λ∪(−Λ)

b|~r |〈~λ, |~r |〉δ〈~λ,~r〉


n∈Ze

2πiλ


λδkλ

We havec0 = λ/2bk = 1/k , k ∈ Z∗

V. Muñoz UCM



Explicit formulas

For the Riemman zeta function,∑γ>0

eiγ|t | + e−|t |/2W0(|t |) = c0δ0 − e−|t |/2∑

(δk log p + δ−k log p)

= c0δ0−∑

p−k/2(log p)(δk log p+δ−k log p)

For a test function ϕ, we have∑ϕ(γ)+W0[ϕ] = c0ϕ(0)−

∑p−k/2(log p)(ϕ(k log p)+ϕ(−k log p)),

where W0[ϕ] =∫

e−|t |/2W0(|t |)ϕ(t)dt .

These are known as Explicit formulas.They do not depend on the functional equation.

V. Muñoz UCM



Explicit formulas


eiγ|t | + e−|t |/2W0(|t |) = c0δ0 − e−|t |/2∑


= c0δ0−∑




where W0[ϕ] =∫

e−|t |/2W0(|t |)ϕ(t)dt .


V. Muñoz UCM



Explicit formulas


eiγ|t | + e−|t |/2W0(|t |) = c0δ0 − e−|t |/2∑


= c0δ0−∑




where W0[ϕ] =∫

e−|t |/2W0(|t |)ϕ(t)dt .


V. Muñoz UCM



Explicit formulas


eiγ|t | + e−|t |/2W0(|t |) = c0δ0 − e−|t |/2∑


= c0δ0−∑




where W0[ϕ] =∫

e−|t |/2W0(|t |)ϕ(t)dt .

These are known as Explicit formulas.

They do not depend on the functional equation.

V. Muñoz UCM



Explicit formulas


eiγ|t | + e−|t |/2W0(|t |) = c0δ0 − e−|t |/2∑


= c0δ0−∑




where W0[ϕ] =∫

e−|t |/2W0(|t |)ϕ(t)dt .


V. Muñoz UCM



Other results

Converse: to a Poisson-Newton formula we associate aDirichlet series.

General Poisson-Newton formula for f meromorphic offinite order.The divisor of a Dirichlet series is not contained in aleft-directed cone.Summation formulas: Abel-Plana, Euler-MacLaurin, etc.Explicit formulas in number theory.Selberg Trace formula in geometry.

V. Muñoz UCM



Other results

Converse: to a Poisson-Newton formula we associate aDirichlet series.General Poisson-Newton formula for f meromorphic offinite order.

The divisor of a Dirichlet series is not contained in aleft-directed cone.Summation formulas: Abel-Plana, Euler-MacLaurin, etc.Explicit formulas in number theory.Selberg Trace formula in geometry.

V. Muñoz UCM



Other results

Converse: to a Poisson-Newton formula we associate aDirichlet series.General Poisson-Newton formula for f meromorphic offinite order.The divisor of a Dirichlet series is not contained in aleft-directed cone.

Summation formulas: Abel-Plana, Euler-MacLaurin, etc.Explicit formulas in number theory.Selberg Trace formula in geometry.

V. Muñoz UCM



Other results

Converse: to a Poisson-Newton formula we associate aDirichlet series.General Poisson-Newton formula for f meromorphic offinite order.The divisor of a Dirichlet series is not contained in aleft-directed cone.Summation formulas: Abel-Plana, Euler-MacLaurin, etc.

Explicit formulas in number theory.Selberg Trace formula in geometry.

V. Muñoz UCM



Other results

Converse: to a Poisson-Newton formula we associate aDirichlet series.General Poisson-Newton formula for f meromorphic offinite order.The divisor of a Dirichlet series is not contained in aleft-directed cone.Summation formulas: Abel-Plana, Euler-MacLaurin, etc.Explicit formulas in number theory.

Selberg Trace formula in geometry.

V. Muñoz UCM



Other results

Converse: to a Poisson-Newton formula we associate aDirichlet series.General Poisson-Newton formula for f meromorphic offinite order.The divisor of a Dirichlet series is not contained in aleft-directed cone.Summation formulas: Abel-Plana, Euler-MacLaurin, etc.Explicit formulas in number theory.Selberg Trace formula in geometry.

V. Muñoz UCM


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Poisson-Newton formulas and Dirichlet series - matematicas… · Classical Poisson formulaDirichlet seriesPoisson formulas for Dirichlet seriesProof of TheoremFurther results Poisson-Newton