Zeta

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

Values at fixed points

Brychkov Yu.A. (2006)

Values at infinities

General characteristics

Domain and analyticity

ζ(s) is an analytical function of s, which is defined over the whole complex s-plane.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

The function ζ(s) has only two singular points:
a) s==1 is the simple pole with residue 1;
b) s==(OverTilde[∞]) is an essential singular point.

Branch points

The function ζ(s) does not have branch points.

Branch cuts

The function ζ(s) does not have branch cuts.

Series representations

Generalized power series

Expansions at s==(Subscript[s, 0])/; (Subscript[s, 0])≠1

Expansions at s==0

Expansions at s==1

Krzysztof Maslanka

Krzysztof Maslanka

Expansions at s==-2n

Exponential Fourier series

Asymptotic series expansions

Other series representations

Krzysztof Maslanka: Hypergeometric-like Representation of the Zeta-Function of Riemann  math-ph/0105007 (2001)
http://arXiv.org/abs/math-ph/0105007

Krzysztof Maslanka: {}[[1,1]] Acta Cosmologica XIII-1, {}[[1,1]] (1997)

For specialized values

G.Huvent (2006)

Integral representations

On the real axis

Of the direct function

For specific values

Multiple integral representations

Product representations

Limit representations

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

The zeta function does not satisfy any algebraic differential equation (D. Hilbert, 1900).

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Identities

Functional identities

For the function itself

Krzysztof Maslanka: Hypergeometric-like Representation of the Zeta-Function of Riemann  math-ph/0105007 (2001)
http://arXiv.org/abs/math-ph/0105007

Including derivatives of the function

Differentiation

Low-order differentiation

General case

Derivatives at zero

Derivatives at other points

Symbolic differentiation

General case

Derivatives at special points

Fractional integro-differentiation

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Definite integration

A. Ivić: Some Identities for the Riemann Zeta Function  math.NT/0305219 (2003)
http://arXiv.org/abs/math.NT/0305219

A. Ivić: Some Identities for the Riemann Zeta Function  math.NT/0305219 (2003)
http://arXiv.org/abs/math.NT/0305219

Summation

Infinite summation

G.Huvent (2006)

Operations

Limit operation

Representations through more general functions

Through hypergeometric functions

Involving HypergeometricPFQ

Through Meijer G

Classical cases for the direct function itself

Through other functions

Representations through equivalent functions

With related functions

Zeros

Sums over zeros

Theorems

The Riemann hypothesis on the zeros of the zeta-function

All nontrivial zeros of ζ(s) lie on the straight line Re(s)==(1/2).

The equivalent version of the Riemann hypothesis

The Riemann hypothesis is equivalent to (∑_ρ)(1/(|ρ((|)^2)))==2+γ-log(4π)/;ζ(ρ)==0∧Im(ρ)≠0 .

A generalization of this result due to Li, Bombieri, Lagarias is:

(1/((n-1)!))((∂^n)/(∂ (s^n)))((s^(n-1))ln((1/2)s(s-1)(π^(-s/2))Γ((s/2))ζ(s)))(Subscript[|, s=1])=(∑_ρ)1-(((1-(1/ρ)))^n)>0 ∀n∈(SuperPlus[ℕ])

A. Weil's "explicit formula"

Let α be any function from (Subscript[C, 0]^∞)(ℝ), Φ(s)=(∫_(-∞)^∞)α(t)exp(s t)dt. Then the following identity holds (the ρ -sum runs over all nontrivial zeros of ζ(z))

Φ(0)+Φ(1)-(∑_ρ)Φ(ρ)==(∑_j^∞)log((Subscript[p, j]))(∑_(k=1)^∞)α(k log((Subscript[p, j])))+(∑_j^∞)log((Subscript[p, j]))(∑_(k=-∞)^(-1))(Subscript[p, j]^k)α(k log((Subscript[p, j])))+α(0)log(π)+(∫_0^∞)(((α(t)-(e^(-t))α(-t))/(1-(e^(-2t))))-α(0)((e^(-2t))/t))dt

The distribution of the zeros

The sequence of zeros of ζ(z) along the critical line 1/2+i t is homogeneously distributed mod 1.

Zeta function regularization

If (∑_(k=1)^∞)(e^(-(Subscript[a, k])z)) is defined for Re(z)>0 and can be analytically continued to a domain containing z==0, then

(lim_(z->0))(∑_(k=1)^∞)(Subscript[a, k])(e^(-(Subscript[a, k])z))==(lim_(s->-1))(∑_(k=1)^∞)(Subscript[a, k]^(-s))

One map with zeta function

The map h : ℝ⟶(ℂ^n) defined by h(t)={ζ(σ+i t),  ζ'(σ+i t), ζ''(σ+i t),…,(ζ^((n)))(σ+i t) } with constant  1/2 < σ < 1  is dense in (ℂ^n).

One max-property

If f(z) is any nonvanishing continous analytic function in the disk |z|<1/4, then there exists a real  t(ε) such that  (max_(|z|<1/4))|ζ(z+(3/4)+i t(ε))-f(z)|<ε.

Montgomery conjecture

The two-point correlation function (Subscript[R, 2])(r)  for the zeros of ζ(z) on the critical line is (Subscript[R, 2])(r)==1-((sin(π r))^2)/(((π r))^2).

Keating–Snaith conjecture

For k∈ℝ, k>-1/2 the following is conjectured:

(lim_(T->∞))(∫_0^T)((|ζ((1/2)+i t)|)^(2k))dt∝(((G(k+1))^2)/(G(2k+1)))a(k)(((log((T/(2π)))))^(k^2))

Here G(z) is the Barnes G function and a(k)=(∏_(n=1)^∞)(((1-(Subscript[p, n])))^(k^2))(∑_(j=0)^∞)(((Γ(j+k)/(j!Γ(k))))^2)(Subscript[p, n]^(-j)).

Hughes–Keating–O’Connel conjecture

For k∈ℝ, k>-3/2 the following is conjectured:

(lim_(T->∞))(∑_(0<(Subscript[s, n])<T))((|ζ'((1/2)+i (Subscript[s, n]))|)^(2k))dt∝(((G(k+2))^2)/(G(2k+3)))a(k)(((log((T/(2π)))))^(k(k+2)))

Here the sum extends over all zeros on the critical line and G(z) is the Barnes G function and a(k)=(∏_(n=1)^∞)(((1-(Subscript[p, n])))^(k^2))(∑_(j=0)^∞)(((Γ(j+k)/(j!Γ(k))))^2)(Subscript[p, n]^(-j)).

GUE hypothesis

A fixed set of  nontrivial zeros of ζ(z) behaves asymptotically like the eigenvalues of a Gaussian unitary ensemble.

Asymptotical behaviour of zeros

The number  N(t) of  zeros (1/2)+i (Subscript[γ, n]) of ζ((1/2)+i t) behaves asymptotically as N(t)==(t/(2π))log((t/(2π e)))+O(log(t)).

The lattice-packing density for any convex symmetrical body

The lattice-packing density (Subscript[δ, n])(𝒦) for any convex symmetrical body 𝒦 in n dimensions satisfies the inequality (Subscript[δ, n])(𝒦)≥((n ζ(n))/(e(1-(e^(-n)))(2^(n-1)))).

The probability of a lattice point to be visible

The probability of a lattice point from (ℤ^d) being visible from the origin (i.e., an d–tuple of integers is relatively prime) is (1/(ζ(d))).

The scattering matrix

The scattering matrix 𝒮 of the Laplace–Beltrami operator in the modular domain has the form 𝒮(ω)=((Γ(1/2) Γ(i ω))/(Γ(1/2+i ω)))((ζ(2i ω))/(ζ(1+2i ω))) .

History

– L. Euler (1737)
– P. G. L. Dirichlet
– P. L. Chebyshev
– B. Riemann (1859)
– J. Hadamard (1893)
– H. von  Mangoldt (1894)
– Ch. J. de la Vallee–Poussin (1896)

Applications include number theory, Bose–Einstein and Fermi–Dirac statistics, analytic approximation and evaluation of integrals and products, regularization techniques in quantum field theory, Scharnhorst effect of quantum electrodynamics, Brownian motion.

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