Fractal Strings, Complex Dimensions and the...

Fractal Strings, Complex Dimensions and theSpectral Operator:

From the Riemann Hypothesis to PhaseTransitions and Universality

Michel L. Lapidus

Department of MathematicsUniversity of California, Riverside

[email protected]://www.math.ucr.edu/~lapidus

New results: Joint work with Hafedh Herichi.

Background material: In collaboration with Machiel van Frankenhuijsen (HelmutMaier and/or Carl Pomerance).

Hawaii Conference in Number Theory

Honolulu, March 7, 2012.

March 6, 2012

Contents

I Background MaterialGeneralized Fractal StringsComplex DimensionsInverse Spectral Problem for Fractal Strings and the RiemannHypothesis (RH)Heuristic Definition/Properties of the Spectral OperatorOperator-Valued Euler Product

II The Spectral OperatorPrecise Definition, Main PropertiesSpectrum: Infinitesimal Shift/Spectral OperatorEvolution Semigroup

III Quasi-Invertibility of the Spectral OperatorTruncated Spectral OperatorsPrecise Reformulation of RHAlmost Invertibility and “Almost RH”

Contents

IV Zeta Values and UniversalitySpectrum of the Spectral Operator, RevisitedOpen Problems

V Spectra, Riemann Zeroes and Phase TransitionsPhase Transitions at c = 1/2 and c = 1

VI Universality of the Spectral OperatorUniversality of the Riemann Zeta FunctionUniversality of the Spectral Operator

Contents

VII Open Problems and Future DirectionsOther L-FunctionsGlobal Zeta Function and Global Spectral OperatorAdelic VersionEuler Product in the Critical Strip (work in progress)Towards a Polya-Hilbert Operator and a Complex/FractalCohomology (ML, 1/12)Special Case of Curves/Varieties over a Finite Field; WeilConjectures, Revisited (ML, 1/12)

VIII Bibliography

Generalized Fractal Strings

Definition

A generalized fractal string η is a local positive or complexmeasure on (0,+∞) satisfying |η|(0, x0) = 0, for some x0 > 0,where |η| is the total variation measure of A defined by

|η|(A) = sup

∞∑k=1

|η(Ak)|

and Ak∞k=1 ranges over all finite or countable partitions of A intomeasurable subsets of (0,+∞).

By “local measure”here, we mean that η is a set function on theBorel class of (0,+∞) (with values in [0,+∞] or C, respectively)whose restriction to any bounded Borel subset of (0,+∞) is apositive or complex measure, respectively.

Definition

|η|(A) = sup

∞∑k=1

|η(Ak)|

Definition

|η|(A) = sup

∞∑k=1

|η(Ak)|

Example

Consider the string consisting of the sequence of lengthsL = a−j∞j=1, with typically nonintegers multiplicities

wj = bj , where 1 < b < a.

Then, the measure associated to this string, called thegeneralized Cantor string (see [La-vF3, §10.1]) is given by

ηCS =∞∑j=0

bjδa−j.

Here, δx is the Dirac measure concentrated at x .

Example

Let Ω be a bounded open subset of R. Write Ω = ∪∞j=1Ij as adisjoint union of (bounded, open) intervals Ij of lengths `j(repeated according to their multiplicity), and such that `j 0 asj → 0. Then Ω (or L := `j∞j=1) is called an ordinary fractalstring. Furthermore,

ηL :=∞∑j=1

δ`−1j

is the associated generalized fractal string.

Definition

Let η be a generalized fractal string. Then its dimension,denoted by Dη, is the abscissa of convergence of the Dirichlet

integral∫ +∞0 x−sη(dx); that is,

Dη = inf

σ ∈ R :

∫ +∞

0x−σ|η|(dx) <∞

Definition

Given a generalized fractal string η, its geometric zetafunction, denoted by ζη, is its Mellin transform; namely,

ζη(s) =

∫ +∞

0x−sη(dx) for Re(s) > Dη .

Definition

Let η be a generalized fractal string. Then its dimension,denoted by Dη, is the abscissa of convergence of the Dirichlet

integral∫ +∞0 x−sη(dx); that is,

Dη = inf

σ ∈ R :

∫ +∞

0x−σ|η|(dx) <∞

Definition

Given a generalized fractal string η, its geometric zetafunction, denoted by ζη, is its Mellin transform; namely,

ζη(s) =

∫ +∞

0x−sη(dx) for Re(s) > Dη .

Definition

We will be interested in the meromorphic continuation ofs 7→ ζη(s). For this purpose, we define the screen as the curveS : S(t) + it, t ∈ R, and the windowW as the subset of thecomplex planeW := s ∈ C : Re(s) ≥ S(Ims).

We assume that s 7→ ζη(s) has a meromorphic continuation tosome neighborhood of W, and we define the set of visible complexdimensions of η as

Dη(W) = ω ∈ W : ζη has a pole at ω .

Example

Harmonic string:

h =∞∑j=1

δj−1

Prime harmonic string:

hp =∞∑j=1

δp−j,

where p ∈ P:= the set of all primes.

Note that h and hp are both generalized fractal strings.Moreover, for any p ∈ P, we have

h = ∗p∈P

where ∗ is the multiplicative convolution of measures on (0,+∞).

Example

Harmonic string:

h =∞∑j=1

δj−1

hp =∞∑j=1

δp−j,

h = ∗p∈P

Example

Harmonic string:

h =∞∑j=1

δj−1

hp =∞∑j=1

δp−j,

h = ∗p∈P

Let η and η′ be two generalized fractal strings. Then we have

ζη∗η′(s) = ζη(s).ζη′(s).

The geometric zeta function associated to hp is

ζhp(s) =1

1− p−s,

the pth Euler factor of ζ(s).

Hence, we have for Re(s) > 1,

ζh(s) = ζ∗hpp∈P

(s) = ζ(s) =∏p∈P

1− p−s=∏p∈P

ζhp(s).

We thus recover the well-known Euler product for ζ.

ζhp(s) =1

1− p−s,

(s) = ζ(s) =∏p∈P

1− p−s=∏p∈P

ζhp(s).

ζhp(s) =1

1− p−s,

(s) = ζ(s) =∏p∈P

1− p−s=∏p∈P

ζhp(s).

Definition

Given a generalized fractal string η, the spectral measure νassociated to η is defined by

ν(A) =+∞∑k=1

for each bounded Borel subset A ⊂ (0,+∞).

We define the spectral zeta function of η to be the geometriczeta function associated to ν; we denote it by ζν .

Let η be a generalized fractal string. Then the spectral measureassociated to η is the convolution of h (the harmonic string) withη. That is,

ν = η ∗ h.

Definition

Given a generalized fractal string η, the spectral measure νassociated to η is defined by

ν(A) =+∞∑k=1

for each bounded Borel subset A ⊂ (0,+∞).

We define the spectral zeta function of η to be the geometriczeta function associated to ν; we denote it by ζν .

Let η be a generalized fractal string. Then the spectral measureassociated to η is the convolution of h (the harmonic string) withη. That is,

ν = η ∗ h.

Theorem

The spectral zeta function of η, denoted by ζν , is obtained bymultiplying ζη by the Riemann zeta function

ζν(s) = ζη(s).ζ(s).

The Distributional Explicit Formulas

Theorem

(See [La-vF3, Ch.5]). Let η be a languid generalized fractalstring. Then, for any k ∈ Z, its kth distributional primitive (or

anti-derivative) P [k]η is given by

P [k]η (x) =

∑ω∈Dη(W)

(x s+k−1ζη(s)

(s)k;ω

(k − 1)!

k−1∑j=0

C k−1j (−1)jxk−1−jζη(−j) +R[k]

η (x),

where R[k]η (x) = 1

∫S x s+k−1ζη(s) ds

(s)kis the error term and can

be suitably estimated as x 7→ +∞.In addition, if η is strongly languid, then we may choose

W = C and Rη(x) ≡ 0.

The Distributional Explicit Formulas

Theorem

(See [La-vF3, Ch.5]). Let η be a languid generalized fractalstring. Then, for any k ∈ Z, its kth distributional primitive (or

anti-derivative) P [k]η is given by

P [k]η (x) =

∑ω∈Dη(W)

(x s+k−1ζη(s)

(s)k;ω

(k − 1)!

k−1∑j=0

C k−1j (−1)jxk−1−jζη(−j) +R[k]

η (x),

where R[k]η (x) = 1

∫S x s+k−1ζη(s) ds

(s)kis the error term and can

be suitably estimated as x 7→ +∞.In addition, if η is strongly languid, then we may choose

W = C and Rη(x) ≡ 0.

When we apply the distributional explicit formulas at levelk = 0, assuming that η is a languid generalized fractal string whosecomplex dimensions are simple and satisfies certain mild additionalconditions, we obtain that, as a distribution, the measure η is givenby the following density of geometric states (or density oflengths) formula:

η =∑

ω∈Dη(W)

res(ζη(s);ω)xω−1.

We also obtain for the spectral measure, by applying the

previous theorem to ν = P [0]ν , the following density of spectral

states (or density of frequencies) formula:

ν = ζη(1) +∑

ω∈Dη(W)

res(ζη(s)ζ(s)x s−1;ω)xω−1

= ζη(1) +∑

ω∈Dη(W)

ζ(ω)res(ζη(s);ω)xω−1,

for simple poles.

The Inverse Spectral Problem

The inverse problem for fractal strings considered in[LaMa1, LaMa2] (and later revisited in [La-vF2, La-vF3]) is thefollowing, for any fixed D ∈ (0, 1):

(IVS)D Given any fractal string L of dimension D such that forsome constants CD and δ > 0,

Nν(x) = W (x)− cDxD + O(xD−δ),

as x → +∞, is it true that L is Minkowski measurable?

(Here, the leading term W (x) is the Weyl term, given byW (x) := vol1(L)x , and below, M is the Minkowski content of L.)

Remark:It follows from results in [LaPo2] and [La2, La3] that if such a

nonzero constant cD exists, then cD > 0 and is given by

cD = 2−(1−D)(1− D)(−ζ(D))M.

where M is the Minkowski content of L.

cD = 2−(1−D)(1− D)(−ζ(D))M.

As a consequence, the main result of [LaMa2] can be stated asfollows:

For any given D ∈ (0, 1), the inverse spectral problem (IVS)Dhas an affirmative answer if and only if ζ(s) 6= 0 for all s ∈ C suchthat Re(s) = D. Hence, (IVS)D is not true in the “mid-fractal”casewhen D = 1

2 , and it holds everywhere else (i.e, for everyD ∈ (0, 1), D 6= 1

2) if and only if the Riemann hypothesis is true.

This spectral reformulation was revisited in [La-vF2, La-vF3] byusing the then rigorously developed theory of complex dimensionsand the associated explicit formulas. (See [La-vF3,Ch.9].)

The Multiplicative and Additive Spectral Operators

Using the distributional explicit formula as a motivation, thespectral operator will be defined at level k = 0 as the map

η 7−→ ν

and at level k = 1 as the map

Nη(x) 7−→ ν(Nη)(x) = Nν(x) =∞∑k=1

η 7−→ ν

Nη(x) 7−→ ν(Nη)(x) = Nν(x) =∞∑k=1

η 7−→ ν

Nη(x) 7−→ ν(Nη)(x) = Nν(x) =∞∑k=1

For every prime number p, we also define the p-factor of ν by

Nη(x) 7−→ νp(Nη)(x) = Nνp(x) = Nη∗hp(x) =∞∑k=0

Nη(xp−k),

where the terms in the sum necessarily vanish when pk ≥ x .

The operators νp commute with each other and theircomposition gives the Euler product for ν :

Nη(x) 7−→ ν(Nη)(x) =( ∏

p∈Pνp

)(Nη).

For every prime number p, we also define the p-factor of ν by

Nη(x) 7−→ νp(Nη)(x) = Nνp(x) = Nη∗hp(x) =∞∑k=0

Nη(xp−k),

where the terms in the sum necessarily vanish when pk ≥ x .

The operators νp commute with each other and theircomposition gives the Euler product for ν :

Nη(x) 7−→ ν(Nη)(x) =( ∏

p∈Pνp

)(Nη).

Making the change of variable x = et (x > 0) orequivalently, t = log x (and hence, t ∈ R), and writingf (t) = Nη(x), we obtain the additive version of the spectraloperator

f (t) 7→ a(f )(t) =∞∑k=1

f (t − log k),

and of its operator-valued Euler factors (for each prime p ∈ P)

f (t) 7→ ap(f )(t) =∞∑k=0

f (t − k log p).

Making the change of variable x = et (x > 0) orequivalently, t = log x (and hence, t ∈ R), and writingf (t) = Nη(x), we obtain the additive version of the spectraloperator

f (t) 7→ a(f )(t) =∞∑k=1

f (t − log k),

and of its operator-valued Euler factors (for each prime p ∈ P)

f (t) 7→ ap(f )(t) =∞∑k=0

f (t − k log p).

The spectral operator a and its Euler factors ap are also relatedby an operator-valued Euler product

f (t) 7−→ a(f )(t) =( ∏

p∈Pap

)(f )(t),

where the product is given in the sense of the composition ofoperators.

If we denote by ∂ := ddt the first order differential operator with

respect to t, the Taylor series associated to f, a smooth function,can be written as

f (t + h) = f (t) +hf ′(t)

h2f′′

2!+ ...

= ehddt (f )(t) = eh∂(f )(t);

that is, ∂ = ddt is the infinitesimal generator of the (one-parameter)

group of shifts on the real line.

This gives a new representation for the spectral operator and itsprime factors:

a(f )(t) =∞∑k=1

e−(log k)∂(f )(t)

=∞∑k=1

)(f )(t)

= ζ(∂)(f )(t)

=∏p∈P

(1− p−∂)−1(f )(t),

and for any prime p,

ap(f )(t) =∞∑k=0

f (t − k log p)

=∞∑k=0

e−k(log p)∂(f )(t)

=∞∑k=0

(p−k∂

)(f )(t)

= (1− p−∂)−1(f )(t)

= ζhp(∂)(t).

The Weighted Hilbert Space Hc

For c ≥ 0, let

f ∈ C∞(R) : supp(f ) ⊂ (0,+∞) and

∫ +∞

0|f (t)|2e−2ctdt <∞

Hc is a pre-Hilbert space for the natural inner product indicatedbelow.

Its completion is a Hilbert space and is denoted by Hc . It isequipped with the following inner product

< f , g >c =

∫ ∞0

f (t)g(t)e−2ctdt

and the associated Hilbert norm ||.||c =√< . , . >c (so that

||f ||2c =∫ +∞0 |f (t)|2e−2ctdt).

For c ≥ 0, let

f ∈ C∞(R) : supp(f ) ⊂ (0,+∞) and

∫ +∞

0|f (t)|2e−2ctdt <∞

< f , g >c =

∫ ∞0

f (t)g(t)e−2ctdt

||f ||2c =∫ +∞0 |f (t)|2e−2ctdt).

For c ≥ 0, let

f ∈ C∞(R) : supp(f ) ⊂ (0,+∞) and

∫ +∞

0|f (t)|2e−2ctdt <∞

< f , g >c =

∫ ∞0

f (t)g(t)e−2ctdt

||f ||2c =∫ +∞0 |f (t)|2e−2ctdt).

The Differentiation Operator A = ∂c

Given c ≥ 0, we define A := ∂ = ∂c = ddt as the unbounded

linear operator from Hc to itself with domain D(A) consisting of allthe functions f ∈ Hc that are (locally) absolutely continuous on R(i.e., f ∈ Cabs(R)) and such that f ′ ∈ Hc (where f’ denotes thepointwise derivative of f, which exists Lebesgue almost everywhereon R). Furthermore, for f ∈ D(A), we let

Af = ∂f := f ′, for all f ∈ D(A).

It follows from the definition of Hc , D(A) and a well-knownlemma (about absolutely continuous functions) that everyf ∈ D(A) naturally satisfies the following boundary conditions at−∞ and +∞, respectively:

(1) (Boundary condition at -∞) f (t) = 0 for all t ≤ 0; inparticular, we have f (0) = 0.

(2) (Boundary condition at +∞) limt→+∞

f (t)e−tc = 0.

Remark:

Intuitively, condition (2) means that the corresponding fractalstrings have (Minkowski) dimension D ≤ c . (See [LapPo2].)

f (t)e−tc = 0.

Remark:

f (t)e−tc = 0.

Remark:

f (t)e−tc = 0.

Remark:

Normality of the unbounded operator ∂c

Theorem

For every c ≥ 0, A = ∂c is an unbounded normal linear operatoron Hc .

Moreover, its adjoint A∗ is given by A∗ = 2c − A, withD(A∗) = D(A).

Normality of the unbounded operator ∂c

Theorem

For every c ≥ 0, A = ∂c is an unbounded normal linear operatoron Hc .

Moreover, its adjoint A∗ is given by A∗ = 2c − A, withD(A∗) = D(A).

Spectrum of ∂c

Theorem

For every c ≥ 0, the spectrum σ(A) of the differentiationoperator A = ∂c is the closed vertical line of the complex planepassing through c. Furthermore, it is equal to the essentialspectrum σe(A) of A:

σ(A) = σe(A) = λ ∈ C|Re(λ) = c .

Moreover, the point spectrum of A is empty (i.e., A does nothave any eigenvalues).

Spectrum of ∂c

Theorem

For every c ≥ 0, the spectrum σ(A) of the differentiationoperator A = ∂c is the closed vertical line of the complex planepassing through c. Furthermore, it is equal to the essentialspectrum σe(A) of A:

σ(A) = σe(A) = λ ∈ C|Re(λ) = c .

Moreover, the point spectrum of A is empty (i.e., A does nothave any eigenvalues).

The strongly continuous group e−t∂c

For every c ≥ 0, e−t∂ct∈R is a strongly continuous group of(bounded linear) operators and

||e−t∂c || = e−ct

for any t ∈ R.

Moreover, its adjoint group (e−t∂c )∗t∈R is given by

e−t∂∗c t∈R = e−t(2c−∂c )t∈R = e−2ctet∂ct∈R.

The strongly continuous group e−t∂c

For every c ≥ 0, e−t∂ct∈R is a strongly continuous group of(bounded linear) operators and

||e−t∂c || = e−ct

for any t ∈ R.

Moreover, its adjoint group (e−t∂c )∗t∈R is given by

e−t∂∗c t∈R = e−t(2c−∂c )t∈R = e−2ctet∂ct∈R.

The strongly continuous group of operators e−t∂t∈R is atranslation (or shift) group. That is, for every t ∈ R,

(e−t∂)(f )(u) = f (u − t)

for all f ∈ Hc and u ∈ R. (For a fixed t ∈ R, this equality holds forelements in Hc and hence, for a.e. u ∈ R.)

Remark: As a result, ∂ = ∂c , the infinitesimal generator of theshift group e−t∂t∈R, is called the infinitesimal shift of the realline (with parameter c ≥ 0).

The strongly continuous group of operators e−t∂t∈R is atranslation (or shift) group. That is, for every t ∈ R,

(e−t∂)(f )(u) = f (u − t)

for all f ∈ Hc and u ∈ R. (For a fixed t ∈ R, this equality holds forelements in Hc and hence, for a.e. u ∈ R.)

Remark: As a result, ∂ = ∂c , the infinitesimal generator of theshift group e−t∂t∈R, is called the infinitesimal shift of the realline (with parameter c ≥ 0).

The Spectrum of a

We can now define the spectral operator as follows:

a = ζ(∂),

via the measurable functional calculus for unbounded normaloperators. (If, for simplicity, we assume c 6= 1 in order to avoid thepole of ζ at s = 1, then ζ is holomorphic in a neighborhood ofσ(∂). If c = 1 is allowed, then we may also use the meromorphicfunctional calculus for sectorial operators; see [Haase].)

Theorem

Assume that c > 1. Then, for any f ∈ D(a), we have

a(f )(t) =∞∑k=1

f (t − log k) = ζ(∂)(f )(t) =∞∑n=1

n−∂(f )(t).

The Spectrum of a

a = ζ(∂),

Theorem

a(f )(t) =∞∑k=1

f (t − log k) = ζ(∂)(f )(t) =∞∑n=1

n−∂(f )(t).

The Spectrum of a

a = ζ(∂),

Theorem

a(f )(t) =∞∑k=1

f (t − log k) = ζ(∂)(f )(t) =∞∑n=1

n−∂(f )(t).

Remark:

For any c > 0, we also show that the above equation holds forall f in a suitable dense subspace of D(a).

Theorem

Assume that c ≥ 0. Then

σ(a) = ζ(σ(∂)) = cl (ζ(λ ∈ C|Re(λ) = c)) ,

where σ(a) is the spectrum of a and N = cl(N) is the closure ofN ⊂ C.

Remark:

For any c > 0, we also show that the above equation holds forall f in a suitable dense subspace of D(a).

Theorem

Assume that c ≥ 0. Then

σ(a) = ζ(σ(∂)) = cl (ζ(λ ∈ C|Re(λ) = c)) ,

where σ(a) is the spectrum of a and N = cl(N) is the closure ofN ⊂ C.

Quasi-Invertibility of the Spectral Operator and a SpectralReformulation of RH

Theorem

Assume that c ≥ 0. Then, the spectral operator a = ζ(∂) isquasi-invertible if and only if the Riemann zeta function does notvanish on the vertical line s ∈ C : Re(s) = c.

Quasi-Invertibility of the Spectral Operator and a SpectralReformulation of RH

Theorem

Assume that c ≥ 0. Then, the spectral operator a = ζ(∂) isquasi-invertible if and only if the Riemann zeta function does notvanish on the vertical line s ∈ C : Re(s) = c.

Corollary

The spectral operator a is quasi-invertible for allc ∈ (0, 1)− 1

2 if and only if the Riemann hypothesis is true.

Remarks:

The notion of quasi-invertibility will be defined in the nextpart.

It suffices to require c ∈ (0, 12) (or c ∈ (12 , 1)) in the abovecorollary. This follows from the functional equation for ζconnecting ζ(s) and ζ(1− s).

Corollary

Remarks:

Corollary

Remarks:

Truncated Spectral Operators and Quasi-Invertibility

We show in [HerLa1] that for any c ≥ 0, the infinitesimal shift∂ = ∂c is given by

∂ = c + iV ,

where V is an unbounded self-adjoint operator such thatσ(V ) = R. Thus, given T ≥ 0, we define the truncatedinfinitesimal shift as follows:

A(T ) = ∂(T ) := c + iV (T ),

whereV (T ) := φ(T )(V )

(in the sense of the functional calculus),

and φ(T ) is a suitable (i.e., T -admissible) continuous (if c 6= 1) ormeromorphic (if c = 1) cut-off function (chosen so that

φ(T )(R) = σ(A(T )

)= c + i [−T ,T ]).

Similarly, the truncated spectral operator is defined (also forc ≥ 0) by

a(T ) := ζ(∂(T )

and φ(T ) is a suitable (i.e., T -admissible) continuous (if c 6= 1) ormeromorphic (if c = 1) cut-off function (chosen so that

φ(T )(R) = σ(A(T )

)= c + i [−T ,T ]).

Similarly, the truncated spectral operator is defined (also forc ≥ 0) by

a(T ) := ζ(∂(T )

More precisely, the T -admissible function φ(T ) is chosen asfollows:

(i) If c 6= 1, φ(T ) is any continuous function such that

φ(T )(R) = [−T ,T ]. (For example, φ(T )(τ) = τ for 0 ≤ τ ≤ T and= T for τ ≥ T ; also, φ(T ) is odd.)

(ii) If c = 1 (which corresponds to the pole of ζ(s) at s = 1),then φ(T ) is a suitable meromorphic analog of (i). (For example,

φ(T )(s) = 2Tπ tan−1(s), so that φ(T )(R) = [−T ,T ].)

One then uses the measurable functional calculus and anappropriate (continuous or meromorphic, for c 6= 1 or c = 1,respectively) version of the Spectral Mapping Theorem (SMT) forunbounded normal operators in order to define both ∂(T ) anda(T ) = ζ(∂(T )), as well as to determine their spectra.

SMT : σ(ψ(L)) = ψ(σ(L))

if ψ is a continuous (resp., meromorphic) function on σ(L) (resp.,on a neighborhood of σ(L)) and L is an unbounded normaloperator.

Note that for c 6= 1 (resp., c = 1), ∂(T ) and a(T ) are thencontinuous (resp., meromorphic) functions of the normal (andsectorial, see [Haa]) operator ∂. An entirely analogous statement istrue for the spectral operator a = ζ(∂).

One then uses the measurable functional calculus and anappropriate (continuous or meromorphic, for c 6= 1 or c = 1,respectively) version of the Spectral Mapping Theorem (SMT) forunbounded normal operators in order to define both ∂(T ) anda(T ) = ζ(∂(T )), as well as to determine their spectra.

SMT : σ(ψ(L)) = ψ(σ(L))

if ψ is a continuous (resp., meromorphic) function on σ(L) (resp.,on a neighborhood of σ(L)) and L is an unbounded normaloperator.

Note that for c 6= 1 (resp., c = 1), ∂(T ) and a(T ) are thencontinuous (resp., meromorphic) functions of the normal (andsectorial, see [Haa]) operator ∂. An entirely analogous statement istrue for the spectral operator a = ζ(∂).

The above construction can be generalized as follows:

Given 0 ≤ T0 ≤ T , one can define a (T0,T )-admissible cut-offfunction φ(T0,T ) exactly as above, except with [−T ,T ] replacedwith τ ∈ R : T0 ≤ |τ | ≤ T.

Correspondingly, one can define V (T0,T ) = φ(T0,T )(V ),

A(T0,T ) = ∂(T0,T ) := c + iV (T0,T )

anda(T0,T ) = ζ(∂(T0,T )),

where ∂(T0,T ) is the (T0,T )-infinitesimal shift and a(T0,T ) is the(T0,T )-truncated spectral operator.

Remark: Note that for T0 = 0, we recover A(T ) and a(T ).

The above construction can be generalized as follows:

Given 0 ≤ T0 ≤ T , one can define a (T0,T )-admissible cut-offfunction φ(T0,T ) exactly as above, except with [−T ,T ] replacedwith τ ∈ R : T0 ≤ |τ | ≤ T.

Correspondingly, one can define V (T0,T ) = φ(T0,T )(V ),

A(T0,T ) = ∂(T0,T ) := c + iV (T0,T )

anda(T0,T ) = ζ(∂(T0,T )),

where ∂(T0,T ) is the (T0,T )-infinitesimal shift and a(T0,T ) is the(T0,T )-truncated spectral operator.

Remark: Note that for T0 = 0, we recover A(T ) and a(T ).

Definition

The spectral operator a is quasi-invertible if its truncation a(T )

is invertible for all T ≥ 0.

Definition

Similarly, a is almost invertible, if for some T0 ≥ 0, itstruncation a(T0,T ) is invertible for all T ≥ T0.

Remark:In the definition of “almost invertibility”, T0 is allowed to

depend on the parameter c .

quasi-invertible ⇒ almost invertible.

Definition

Theorem

For all T ≥ 0, A(T ) and a(T ) are bounded normal operators,with spectra respectively given by

A(T ))

= c + iτ : |τ | ≤ T

σ(a(T )

)= ζ(c + iτ) : |τ | ≤ T.

Remarks:

Recall that a(T ) is invertible if and only if 0 /∈ σ(a(T )

More generally, given 0 ≤ T0 ≤ T , the exact counterpart ofthe above theorem holds for A(T0,T ) and a(T0,T ), except with|τ | ≤ T replaced with T0 ≤ |τ | ≤ T .

Theorem

A(T ))

= c + iτ : |τ | ≤ T

andσ(a(T )

)= ζ(c + iτ) : |τ | ≤ T.

Remarks:

Theorem

A(T ))

= c + iτ : |τ | ≤ T

andσ(a(T )

)= ζ(c + iτ) : |τ | ≤ T.

Remarks:

Theorem

A(T ))

= c + iτ : |τ | ≤ T

andσ(a(T )

)= ζ(c + iτ) : |τ | ≤ T.

Remarks:

Corollary

Assume that c ≥ 0. Then, the truncated spectral operator a(T )

is invertible if and only if ζ does not vanish on the vertical linesegment s ∈ C : Re(s) = c , |Im(s)| ≤ T.

Remark:Naturally, given 0 ≤ T0 ≤ T , the same result is true for a(T0,T )

provided |Im(s)| ≤ T is replaced with T0 ≤ |Im(s)| ≤ T .

Corollary

Assume that c ≥ 0. Then, the truncated spectral operator a(T )

is invertible if and only if ζ does not vanish on the vertical linesegment s ∈ C : Re(s) = c , |Im(s)| ≤ T.

Remark:Naturally, given 0 ≤ T0 ≤ T , the same result is true for a(T0,T )

provided |Im(s)| ≤ T is replaced with T0 ≤ |Im(s)| ≤ T .

Theorem

Assume that c ≥ 0. Then,

1 a is quasi-invertible if and only if ζ does not vanish (i.e., doesnot have any zeroes) on the vertical line Re(s) = c.

2 a is almost invertible if and only if all but (at most) finitelymany zeroes of ζ are off the vertical line Re(s) = c.

Theorem

Corollary

1 a is quasi-invertible for all c ∈ (12 , 1) if and only if theRiemann hypothesis is true.

2 a is almost invertible for all c ∈ (12 , 1) if and only if theRiemann hypothesis (RH) is “almost true ”(i.e., on everyvertical line Re(s) = c, c > 1

2 , there are at most finitely manyexceptions to RH).

Corollary

1 a is quasi-invertible for all c ∈ (12 , 1) if and only if theRiemann hypothesis is true.

2 a is almost invertible for all c ∈ (12 , 1) if and only if theRiemann hypothesis (RH) is “almost true ”(i.e., on everyvertical line Re(s) = c, c > 1

2 , there are at most finitely manyexceptions to RH).

Remark:According to our previous results, we have (for the spectral

operator a):

1 invertible ⇒ quasi-invertible ⇒ almost invertible.

2 For c = 12 , a is not almost (and hence, not quasi- etc.)

invertible. (This follows from Hardy’s theorem according towhich ζ has infinitely many zeroes on the critical lineRe(s) = 1

3 For c > 1, a is quasi- (and hence, almost) invertible. In fact,we will next see that a is also invertible for c > 1.

operator a):

Spectrum of a Revisited: Zeta Values and Universality

Recall that, by the Spectral Mapping Theorem ( SMT) forunbounded normal operators, σ(a) (the spectrum of a) is equal tothe closure of the range of ζ on the vertical lineσ(A) = Re(s) = c. Hence, σ(a) is equal to ζ(c + iτ) : τ ∈ Runion its limit points (in C).

Also, by definition of σ(a), a is invertible if and only if 0 /∈ σ(a).

Theorem

1 For c > 1, σ(a) is bounded and 0 /∈ σ(a). Hence, a isinvertible.

2 (Universality) For c ∈ (12 , 1), σ(a) = C. (This follows from theBohr–Landau Theorem and, more generally, from theuniversality of ζ in the right critical strip 1

2 < Re(s) < 1.)Hence, a is not invertible (because 0 ∈ σ(a)).

3 For c ∈ (0, 12), σ(a) is unbounded and conditionally (i.e.,under RH), σ(a) 6= C and, in particular, 0 /∈ σ(a), so that a isinvertible.

Remark:The last statement in the third part of the theorem follows from

the non-universality of ζ in the left critical strip 0 < Re(s) < 12 ;

see [KaSt].

Theorem

see [KaSt].

Theorem

see [KaSt].

Theorem

see [KaSt].

Theorem

see [KaSt].

Open Problems

1 What is σ(a) when c ∈ (0, 12)? (It is a very complicated,closed, unbounded, and (conditionally) strict subset of C.)Is RH needed for σ(a) 6= C to be true? (Compare [KaSt].)

2 Does 0 /∈ σ(a), when c ∈ (0, 12)? Unconditionally (or elseunder the Lindelof hypothesis), we conjecture that 0 /∈ σ(a)and hence, that a is invertible for 0 < c < 1

3 Conjecturally, for c = 12 , we have that σ(a) = C. Moreover, a

is clearly not invertible for c = 12 since ζ has zeroes on the

critical line and hence, we know that 0 ∈ σ(a).

Open Problems

Spectra, Riemann Zeroes, and Phase Transitions

I. Phase Transition at c = 12

Theorem

Conditionally (i.e., under RH), the spectral operator a isquasi-invertible for all c 6= 1

2 (c ∈ (0, 1)), and (unconditionally) itis not quasi-invertible (not even almost invertible) for c = 1

Remark:

1 Recall that a is quasi-invertible for all c 6= 12 ⇐⇒ RH.

2 Furthermore, a is not almost invertible for c = 12 because ζ

has infinitely many zeroes on the critical line Re(s) = 12 .

Theorem

Remark:

Theorem

Remark:

Theorem

Remark:

Theorem

Remark:

II. Phase Transitions at c = 12 and c = 1

Theorem

The spectral operator a is invertible for c ≥ 1, is not invertiblefor 1

2 < c < 1 (conjecturally, also for c = 12), and invertible

(conditionally) for 0 < c < 12 .

Theorem

The spectrum σ(a) is non-compact (and hence, unbounded),but (conditionally) not all of C for 0 < c < 1

2 . It is all of C for12 < c < 1, and compact (and thus, bounded) for c > 1.

Theorem

III. Phase Transition at c = 1

Theorem

Unconditionally, the spectral operator a is unbounded for c < 1,and is bounded for c > 1.

III. Phase Transition at c = 1

Theorem

Unconditionally, the spectral operator a is unbounded for c < 1,and is bounded for c > 1.

Universality of the Riemann Zeta Function ζ = ζ(s)

The “universality”of ζ roughly means that any non-vanishingholomorphic function in 12 < Re(s) < 1 can be approximatedarbitrarily closely by imaginary translates of ζ.

More precisely, we have the following well-known remarkableresult (Extended Voronin Theorem).

Theorem

Let K be any compact subset of 12 < Re(s) < 1, withconnected complement in C. Let g : K → C be a non-vanishingcontinuous function that is holomorphic in the interior of K (whichmay be empty). Then, given any ε > 0, there exists τ ≥ 0(depending only on ε) such that

Jsc(τ) := maxs∈K|g(s)− ζ(s + iτ)| ≤ ε.

Theorem

In fact, the set of such τ ’s has a positive lower density and, inparticular, is infinite. More precisely, we have

lim infρ→+∞

ρvol1 (τ ∈ [0, ρ] : Jsc(τ) ≤ ε) > 0.

Remarks:

Voronin’s Original Universality Theorem (1975) correspondsto the choice of K := D(34 , r), the closed disk of center 3

4 andradius r , with 0 < r < 1

4 arbitrary.

The compact set K is allowed to have empty interior, in whichcase f is only required to be continuous (and without zeroes)in K . In particular, if K is a line segment (on the real axis),then any continuous curve can be approximated by imaginarytranslates of ζ. Thus ζ encodes all types of complex behaviors:it is chaotic.

If we assume the Riemann hypothesis, then ζ(s) does not haveany zeroes in 1

2 < Re(s) < 1. Hence, applying the UniversalityTheorem to g(s) := ζ(s) and upon some elementarymanipulations, one sees that scaled copies of ζ can be foundwithin itself at all scales. In other words, conditionally, theRiemann zeta function is both fractal and chaotic.

Remarks:

4 arbitrary.

Remarks:

4 arbitrary.

Universality of the Spectral Operator a = ζ(∂)

The “universality”of the spectral operator a = ζ(∂) roughlymeans that any non-vanishing holomorphic function of ∂ in12 < Re(s) < 1 can be approximated arbitrarily closely byimaginary translates of ζ(∂).

More precisely, we have the following operator-theoreticgeneralization of the Extended Voronin Universality Theorem,expressed in terms of the imaginary translates of the T -truncated

spectral operators a(T ) = ζ(∂(T )), where ∂(T ) = ∂(T )c is the

T -truncated infinitesimal shift (with parameter c).

Universality of the Spectral Operator a = ζ(∂)

The “universality”of the spectral operator a = ζ(∂) roughlymeans that any non-vanishing holomorphic function of ∂ in12 < Re(s) < 1 can be approximated arbitrarily closely byimaginary translates of ζ(∂).

More precisely, we have the following operator-theoreticgeneralization of the Extended Voronin Universality Theorem,expressed in terms of the imaginary translates of the T -truncated

spectral operators a(T ) = ζ(∂(T )), where ∂(T ) = ∂(T )c is the

T -truncated infinitesimal shift (with parameter c).

We begin by providing an operator-theoretic generalization ofthe Universality Theorem that is in the spirit of Voronin’s OriginalUniversality Theorem.

Theorem

Let K be a compact subset of 12 < Re(s) < 1 of the followingform. Assume, for simplicity, that K = K × [−T0,T0], for someT0 ≥ 0, where K is a compact subset of (12 , 1).

Let g : K → C be a non-vanishing continuous function that isholomorphic in the interior of K (which may be empty). Then,given any ε > 0, there exists τ ≥ 0 (depending only on ε) such that

Hop(τ) := supc∈K, 0≤T≤T0

∣∣∣∣∣∣g (∂(T )c

)− ζ

(∂(T )c + iτ

) ∣∣∣∣∣∣ ≤ ε,where ∂(T ) = ∂

(T )c is the T -truncated infinitesimal shift (with

parameter c) and ||.|| is the norm in B(Hc)) (the space of boundedlinear operators on Hc).

Theorem

∣∣∣∣∣∣g (∂(T )c

)− ζ

(∂(T )c + iτ

) ∣∣∣∣∣∣ ≤ ε,where ∂(T ) = ∂

Theorem

∣∣∣∣∣∣g (∂(T )c

)− ζ

(∂(T )c + iτ

) ∣∣∣∣∣∣ ≤ ε,where ∂(T ) = ∂

Theorem

∣∣∣∣∣∣g (∂(T )c

)− ζ

(∂(T )c + iτ

) ∣∣∣∣∣∣ ≤ ε,where ∂(T ) = ∂

In fact, the set of all such τ ’s has a positive lower density and,in particular, is infinite. More precisely, we have

lim infρ→+∞

ρvol1 (τ ∈ [0, ρ] : Hop(τ) ≤ ε) > 0.

Remark:

A remarkable feature of the above generalization is theuniformity in the parameter c ∈ K and in T ∈ [0,T0] of the stated

approximation of g(∂(T )c

In fact, the set of all such τ ’s has a positive lower density and,in particular, is infinite. More precisely, we have

lim infρ→+∞

ρvol1 (τ ∈ [0, ρ] : Hop(τ) ≤ ε) > 0.

Remark:

A remarkable feature of the above generalization is theuniformity in the parameter c ∈ K and in T ∈ [0,T0] of the stated

approximation of g(∂(T )c

We will next state a further generalization of theoperator-theoretic Extended Voronin Universality Theorem. Forpedagogical reasons, we will choose assumptions (on the compactset K ) that simplify its formulation. (The appropriate definitionsand possible extensions will be given just after the statement ofthe theorem.)

Theorem

Let K be any compact, vertically convex, subset of12 < Re(s) < 1, with connected complement in C. Assume, forsimplicity, that K is symmetric with respect to the real axis. Denoteby K the projection of K onto the real axis, and for c ∈ K, let

T (c) := sup (T ≥ 0 : [c − iT , c + iT ] ⊂ K) .

(By construction, K is a compact subset of (12 , 1) and0 ≤ T (c) <∞, for c ∈ K.) Assume further that c 7→ T (c) iscontinuous on K.

Jop(τ) := supc∈K, 0≤T≤T (c)

∣∣∣∣∣∣g (∂(T )c

)− ζ

(∂(T )c + iτ

) ∣∣∣∣∣∣ ≤ ε,

Theorem

T (c) := sup (T ≥ 0 : [c − iT , c + iT ] ⊂ K) .

∣∣∣∣∣∣g (∂(T )c

)− ζ

(∂(T )c + iτ

) ∣∣∣∣∣∣ ≤ ε,

Theorem

T (c) := sup (T ≥ 0 : [c − iT , c + iT ] ⊂ K) .

∣∣∣∣∣∣g (∂(T )c

)− ζ

(∂(T )c + iτ

) ∣∣∣∣∣∣ ≤ ε,

where ∂(T ) = ∂(T )c is the T -truncated infinitesimal shift (with

parameter c) and ||.|| denotes the usual norm in B(Hc) (the spaceof bounded linear operators on Hc).

In fact, the set of such τ ′s has a positive lower density and, inparticular, is infinite. More precisely, we have

lim infρ→+∞

ρvol1(τ ∈ [0, ρ] : Jop(τ) ≤ ε) > 0.

Remarks:

To say that K is vertically convex means that if c − iT ’ andc + iT belong to K for some c ∈ K and T ′ ≤ 0 ≤ T , thenthe entire line segment [c − iT ′, c + iT ] is contained in K .

Instead of assuming that K is symmetric with respect to thereal axis, it would suffice to suppose that c + iT ∈ K (forsome c ∈ K and T > 0) implies that c − iT ∈ K , and viceversa.

As in the scalar case (and taking K to be a line segment), we

see that any continuous curve(

of ∂(T )c

)can be approximated

by imaginary translates of a(T ) = ζ(∂(T )c

). Hence, roughly

speaking, the spectral operator a(

or its T -truncations a(T ))

can emulate any type of complex behavior: it is chaotic.

Remarks:

of ∂(T )c

). Hence, roughly

Remarks:

of ∂(T )c

). Hence, roughly

Remarks:

of ∂(T )c

). Hence, roughly

Conditionally (i.e., under RH), and applying the aboveoperator-theoretic Universality Theorem to g(s) := ζ(s), wesee that, roughly speaking, arbitrarily small scaled copies ofthe spectral operator are encoded within a itself. In otherwords, a (or its T -truncation) is both chaotic and fractal.

Future Research Directions

1 Study of the global spectral operator

a = ξ(∂),

where ξ(s) = π−s2 Γ( s2)ζ(s) is the global Riemann zeta

function.

2 Study of the Euler products representation of a; see[HerLa3]. Adelic representation of a.

3 Extension to other L-functions (e.g., Dirichlet L-funcions, zetafunctions of number fields, etc..), as well as to members ofthe Selberg class.

4 Spectral operator and universality (both for ζ and otherL-functions).

a = ξ(∂),

function.

a = ξ(∂),

function.

a = ξ(∂),

function.

a = ξ(∂),

function.

BIBLIOGRAPHY

H. Brezis, Functional Analysis, Sobolev Spaces and PartialDifferential Equations, Universitext, Springer, New York, 2011.(English transl. and rev. and enl. ed. of H. Brezis, AnalyseFonctionelle: Theorie et applications, Masson, Paris, 1983.)

D. L. Cohn, Measure Theory, Birkhauser, Boston, 1980.

R. Courant and D. Hilbert, Methods of Mathematical Physics,vol.I, English translation, Interscience, New York, 1953.

H. M. Edwards, Riemann’s Zeta Function, Academic Press,New York, 1974.

K. J. Falconer, Fractal Geometry: Mathematical foundationsand applications, John Wiley and Sons, Chichester, 1990.

G. B. Folland, Real Analysis: Modern Techniques and TheirApplications, 2nd. ed., John Wiley & Sons, Boston, 1999.

I. M. Gelfand and G. E. Shilov, Generalized Functions, Vols.I,IIand III, Academic Press, new edition, 1986.

M. Haase, The Functional Calculus for Sectorial Operators,Birkhauser Verlag, Berlin, 2006.

B. M. Hambly and M. L. Lapidus, Random fractal strings:their zeta functions, complex dimensions and spectralasymptotics, Trans. Amer. Math. Soc. No. 1, 358 (2006),285-314.

C. Q. He and M. L. Lapidus, Generalized Minkowski content,spectrum of fractal drums, fractal strings and the Riemannzeta-function, Memoirs Amer. Math. Soc. No. 608, 127(1997), 1-97.

H. Herichi and M. L. Lapidus, Fractal strings, the spectraloperator and the Riemann hypothesis: Zeta values, Riemannzeroes and phase transitions, preprint, 2011.

H. Herichi and M. L. Lapidus, Invertibility of the spectraloperator and a reformulation of the Riemann Hypothesis, inpreparation, 2011.

H. Herichi and M. L. Lapidus, Convergence of the Eulerproduct for the spectral operator in the critical strip, inpreparation, 2011.

A. E. Ingham, The Distribution of Prime Numbers, 2nd ed.(reprinted from the 1932 ed.), Cambridge Univ. Press,Cambridge, 1992.

A. Ivic, The Riemann Zeta-Function: The theory of theRiemann zeta-function with applications, John Wiley andSons, New York, 1985.

G. W. Johnson and M. L. Lapidus, The Feynman Integral andFeynman’s Operational Calculus, Oxford MathematicalMonographs, Oxford Univ. Press, Oxford, 2000.

J. Kalpokas and J. Steuding, On the value-distribution of theRiemann-zeta function on the critical line, preprint, 2009.

A. A. Karatsuba and S. M. Voronin, The RiemannZeta-Function, De Gruyter, Berlin, 1992.

T. Kato, Perturbation Theory for Linear Operators, SpringerVerlag, New York, 1995.

M. L. Lapidus, Fractal drum, inverse spectral problems forelliptic operators and a partial resolution of the Weyl-Berryconjecture, Trans. Amer. Math. Soc. 325 (1991), 465-529.

M. L. Lapidus, Spectral and Fractal Geometry: From theWeyl-Berry conjecture for the vibrations of fractal drums tothe Riemann zeta-function, in: Differential Equations andMathematical Physics (C. Bennewitz, ed.), Proc. Fourth UABIntern. Conf. (Birmingham, March 1990), Academic Press,New York, 1992, pp. 151-182.

M. L. Lapidus,Vibrations of fractal drums, the Riemannhypothesis, waves in fractal media, and the Weyl-Berryconjecture, in: Ordinary and Partial Differential Equations (B.D. Sleeman and R. J. Jarvis, eds.), vol. IV, Proc. TwelfthInternat. Conf. (Dundee, Scotland, UK, June 1992), PitmanResearch Notes in Math. Series, vol. 289, Longman Scientificand Technical, London, 1993, pp. 126-209.

M. L. Lapidus, Fractals and vibrations: Can you hear the shapeof a fractal drum?, Fractals 3, No. 4 (1995), 725-736. (Specialissue in honor of Benoit B. Mandelbrot’s 70th birthday.)

M. L. Lapidus, In Search of the Riemann Zeros: Strings,fractal membranes and noncommutative spacetimes, Amer.Math. Soc., Providence, R.I., 2008.

M. L. Lapidus, J. Levy Vehel and J. A. Rock, Fractal stringsand multifractal zeta functions, Lett. Math. Phys. No. 1, 88

(2009), 101-129. (E-print: arXiv: math-ph/0610015v3, 2009;Springer Open Access: DOI 10.1007/s11005-009-0302-y.)

M. L. Lapidus and H. Lu, Self-similar p-adic fractal strings andtheir complex dimensions, p-Adic Numbers, UltrametricAnalysis and Applications (Russian Academy of Sciences,Moscow) No. 2, 1 (2009), 167-180.

M. L. Lapidus and H. Maier, Hypothese de Riemann, cordesfractales vibrantes et conjecture de Weyl-Berry modifiee, C. R.Acad. Paris Ser. I Math. 313 (1991), 19-24.

M. L. Lapidus and H. Maier, The Riemann hypothesis andinverse spectral problems for fractal strings, J. London Math.Soc. (2). 52(1995), 15-34.

M. L. Lapidus and E. P. J Pearse, Tube formulas and complexdimensions of self-similar tilings, Acta ApplicandaeMathematicae No. 1, 112 (2010), 91-137. (E-print: arXiv:

math.DS/0605527v5; 2010 Springer Open Access: DOI10.1007/S10440-010-9562-x.)

M. L. Lapidus, E. P. J Pearse and S. Winter, Pointwise tubeformulas for fractal sprays and self-similar tilings with arbitrarygenerators, Adv. Math. 227 (2011), 1349-1398. (E-print:arXiv:1006.3807v1 [math.MG], 2010.)

M. L. Lapidus and C. Pomerance, Fonction zeta de Riemannet conjecture de Weyl-Berry pour les tambours fractals, C. R.Acad. Sci. Paris Ser. I Math. 310 (1990), 343-348.

M. L. Lapidus and C. Pomerance, The Riemann zeta-functionand the one-dimensional Weyl-Berry conjecture for fractaldrums, Proc. London Math. Soc. (3)66 (1993), 41-69.

M. L. Lapidus and C. Pomerance, Counterexamples to themodified Weyl-Berry conjecture on fractal drums, Math. Proc.Cambridge Philos. Soc. 119 (1996), 167-178.

M. L. Lapidus and M. van Frankenhuijsen, Complexdimensions of fractal strings and oscillatory phenomena infractal geometry and arithmetic, in : Spectral Problems inGeometry and Arithmetic (T. Branson, ed.), ContemporaryMathematics, vol. 237, Amer. Math. Soc., Providence, R. I.,1999, pp. 87-105.

M. L. Lapidus and M. van Frankenhuijsen, Fractal Geometryand Number Theory (Complex dimensions of fractal stringsand zeros of zeta functions), Birkhauser, Boston, 2000.

M. L. Lapidus and M. van Frankenhuijsen, Fractal Geometry,Complex Dimensions and Zeta Functions: Geometry andspectra of fractal strings, Springer Monographs inMathematics, Springer, New York, 2006. (Second rev. and erl.ed. to appear in 2011.)

A. Laurincikas, Limit Theorems for the RiemannZeta-Function, Kluwer Academic Publishers, Dordrecht, 1996.

B. B. Mandelbrot, The Fractal Geometry of Nature, rev. andenl. ed. (of the 1977 ed.), W. H. Freeman, New York, 1983.

P. Mattila, Geometry of Sets and Measures in EuclideanSpaces (Fractals and Rectifiability), Cambridge Univ. Press,Cambridge, 1995.

S. J. Patterson, An Introduction to the Theory of the RiemannZeta-Function, Cambridge Univ. Press, Cambridge, 1988.

M. Reed and B. Simon, Methods of Modern MathematicalPhysics, vol.I, Functional Analysis, rev. and enl. ed. (of the1975 ed), Academic Press, New York, 1980.

M. Reed and B. Simon, Methods of Mathematical Physics,vols. I-IV, Academic Press, New York, 1979.

W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill,New York, 1987.

W. Rudin, Functional Analysis, 2nd ed. (of the 1973 ed.),McGraw-Hill, New York, 1991.

M. Schechter, Operator Methods in Quantum Mechanics,Dover Publications, 2003.

L. Schwartz, Theorie des Distributions, rev. and enl. ed. (ofthe 1951 ed.), Hermann, Paris, 1996.

L. Schwartz, Methodes Mathematiques pour les SciencesPhysiques, Hermann, Paris, 1961.

J. Steuding, Value-Distribution of L-Functions, Lecture Notesin Mathematics, vol. 1877, Springer, Berlin, 2007.

E. C. Titchmarsh, The Theory of the Riemann Zeta-Function,2nd ed. (revised by D.R. Heath-Brown), Oxford Univ. Press,Oxford, 1986.

Fractal Strings, Complex Dimensions and the...

Documents

Transcript of Fractal Strings, Complex Dimensions and the...