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Summability method in physics
In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent
Zeta_function_regularization
Divergent series
can be assigned a finite value. In particular, the methods of zeta function regularization and Ramanujan summation assign the series a value of −+1/12
1_+_2_+_3_+_4_+_⋯
Divergent series
methods for obtaining values from divergent series, including zeta function regularization. 1 + 1 + 1 + 1 + ⋯ is a divergent series, meaning that its sequence
1_+_1_+_1_+_1_+_⋯
Method in evaluating divergent integrals
In theoretical physics, dimensional regularization is a method for regularizing integrals in the evaluation of Feynman diagrams; in other words, assigning
Dimensional_regularization
Analytic function in mathematics
Riemann zeta function, or Euler–Riemann zeta function, denoted by the lowercase Greek letter ζ {\displaystyle \zeta } (zeta), is a function of a complex
Riemann_zeta_function
Infinite series that is not convergent
then its value at s = −1 is called the zeta regularized sum of the series a1 + a2 + ... Zeta function regularization is nonlinear. In applications, the numbers
Divergent_series
Method used in mathematical physics
regularization Lattice regularization Pauli–Villars regularization Zeldovich regularization Zeta function regularization Perturbative predictions by quantum field
Regularization_(physics)
Mathematical method extending convergence
mathematics, Hadamard regularization (also called Hadamard finite part or Hadamard's partie finie) is a method of regularizing divergent integrals by
Hadamard_regularization
Mathematical function
^{2}}{6\gamma }}\end{aligned}}} also hold true. The digamma function appears in the regularization of divergent integrals ∫ 0 ∞ d x x + a , {\displaystyle
Digamma_function
Regularization technique in quantum field theory
regularization is more difficult to use in QCD calculations. P–V serves as a helpful alternative to the more commonly used dimensional regularization
Pauli–Villars_regularization
Mathematical techniques for summing divergent infinite series
{a^{m-2r+1}}{m-2r+1}}.} Note that this involves (see Renormalization § Zeta function regularization) I ( n , Λ ) = ∫ 0 Λ d x x n . {\displaystyle I(n,\Lambda )=\int
Ramanujan_summation
Differential operator
practice both numbers are often infinite so are defined using zeta function regularization. It was introduced by Atiyah, Patodi, and Singer (1973, 1975)
Eta_invariant
Constants of the mathematical zeta function
Riemann zeta function is a function in complex analysis, which is also important in number theory. It is often denoted ζ ( s ) {\displaystyle \zeta (s)}
Particular values of the Riemann zeta function
Particular_values_of_the_Riemann_zeta_function
Determinant in functional analysis
perform some kind of regularization. The most popular of which for computing functional determinants is the zeta function regularization. For instance, this
Functional_determinant
Expression which is not assigned an interpretation
Singularity - a point at which a mathematical function ceases to be well-behaved Zeta function regularization - in physics, a method for assigning values
Undefined_(mathematics)
Quantum field theory on a lattice
regularization Lattice regularization Zeta function regularization Causal perturbation theory Hadamard regularization Point-splitting regularization v
Lattice_field_theory
Objects extending the notion of functions
first case, the multiplication is determined with some regularization of generalized function. In the second case, the algebra is constructed as multiplication
Generalized_function
Renormalization scheme in quantum field theory
diagram calculations into the counterterms. When using dimensional regularization, i.e. d 4 p → μ 4 − d d d p , {\displaystyle \ \mathrm {d} ^{4}p\to
Minimal_subtraction_scheme
Mathematical series
{S_{f}(y)}{y^{s+1}}}dy.} Z transform General Dirichlet series Zeta function regularization Euler product Dirichlet convolution The formulas for both series
Dirichlet_series
Sigmoid shape special function
mathematics, the error function (also called the Gauss error function), often denoted by e r f {\displaystyle \mathbf {erf} } , is the function erf ( z ) = 2
Error_function
Method in physics used to deal with infinities
the existing loops at large momenta. Yet another regularization scheme is the lattice regularization, which places four-dimensional spacetime on a lattice
Renormalization
Mathematical conjecture about the Riemann zeta function
Hilbert–Pólya conjecture states that the non-trivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator. It is a possible
Hilbert–Pólya_conjecture
Linear transform from the time domain to the frequency domain
Probability-generating function Star transform Zak transform Zeta function regularization Mandal, Jyotsna Kumar (2020). "Z-Transform-Based Reversible Encoding"
Z-transform
Difference between logarithm and harmonic series
derivative of the Riemann zeta function and Dirichlet beta function. In connection to the Laplace and Mellin transform. In the regularization/renormalization of
Euler's_constant
Divergences arising from high energy physics
Renormalization group UV fixed point Causal perturbation theory Zeta function regularization J.D. Bjorken, S. Drell (1965). Relativistic Quantum Fields, Preface
Ultraviolet_divergence
Force resulting from the quantisation of a field
computed using Euler–Maclaurin summation with a regularizing function (e.g., exponential regularization) not so anomalous as |ωn|−s in the above. Casimir's
Casimir_effect
Concept in theoretical physics
computational method based on the function ψ(g) = G d/(∂G/∂g) which they introduced. Like the earlier function h(e), their function determines the change of the
Renormalization_group
Rivoal, Tanguy (2012). "On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant". Michigan Mathematical Journal
List_of_numbers
Renormalization scheme in quantum field theory
where the left-hand side of the equation is the two-point correlation function of the Dirac field. In a new theory, the Dirac field can interact with
On-shell renormalization scheme
On-shell_renormalization_scheme
Mathematically rigorous approach to renormalization theory
ultraviolet divergences in the corresponding calculations. From the generalized functions point of view, the problem of divergences is rooted in the fact that the
Causal_perturbation_theory
Formalism in string theory
field theory String cosmology Supergravity The Elegant Universe Zeta function regularization Sen, Ashoke (1999-12-29). "Universality of the tachyon potential"
String_field_theory
Series of functions in mathematics
remainder terms. These regularization methods compute a series remainder by applying Borel summation or Mellin-Barnes regularization to the series tail.
Asymptotic_expansion
'Best' approximation of a function by a rational function of given order
the function f(x). The zeta regularization value at s = 0 is taken to be the sum of the divergent series. The functional equation for this Padé zeta function
Padé_approximant
German mathematician (born 1958)
of L-functions at both finite and infinite places using regularized determinants. For example, for the Euler factors of the Riemann zeta-function this
Christopher_Deninger
(disambiguation) Zero lift axis Zero sound Zeroth law of thermodynamics Zeta function regularization Zevatron Ze'ev Lev Zhang Jie (scientist) Zhao Jiuzhang Zhores
Index_of_physics_articles_(Z)
Polynomial related to differential operators
1007/BF01076413. ISSN 0016-2663. Sato, Mikio; Shintani, Takuro (June 1972). "On Zeta Functions Associated with Prehomogeneous Vector Spaces". Proceedings of the National
Bernstein–Sato_polynomial
Class of algorithms for solving constrained optimization problems
together with extensions involving non-quadratic regularization functions (e.g., entropic regularization). This combined study gives rise to the "exponential
Augmented_Lagrangian_method
related to the first-order term in Taylor's theorem Hadamard regularization Hadamard's gamma function Hadamard multiplication theorem – for the singularities
List of things named after Jacques Hadamard
List_of_things_named_after_Jacques_Hadamard
Inverse of a finite difference
{\displaystyle \sum x^{a}=-\zeta (-a,x)+C(x).} where ζ ( − a , x ) {\displaystyle \zeta (-a,x)} is the Hurwitz zeta function. Falling factorials: ∑ ( x
Indefinite_sum
Mathematical theorem about the real analytic Eisenstein series
of a real analytic Eisenstein series (or Epstein zeta function) in terms of the Dedekind eta function. There are many generalizations of it to more complicated
Kronecker_limit_formula
Fundamental solution to the heat equation, given boundary values
type. Heat kernel signature Heat kernel regularization Mehler kernel Minakshisundaram–Pleijel zeta function Weierstrass transform § Generalizations Evans
Heat_kernel
Relativistic wave equation describing massless fermions
_{\mu }+\zeta m\omega K\right)=-\left(\partial _{t}^{2}-{\vec {\nabla }}\cdot {\vec {\nabla }}+\eta \zeta ^{*}m^{2}\right)=-\left(\square +\eta \zeta ^{*}m^{2}\right)
Weyl_equation
Infinite series summing alternating 1 and -1 terms
used to sum it are discussed in the articles on regularization and, in particular, the zeta function regulator. The Grandi series has been applied to
Grandi's_series
Topological invariant of manifolds that can distinguish homotopy-equivalent manifolds
eigenvalues on k-forms are λj then the zeta function ζk is defined to be ζ k ( s ) = ∑ λ j > 0 λ j − s {\displaystyle \zeta _{k}(s)=\sum _{\lambda _{j}>0}\lambda
Analytic_torsion
Functional relationship between two quantities
}}}} where ζ ( α , x m i n ) {\displaystyle \zeta (\alpha ,x_{\mathrm {min} })} is the incomplete zeta function. The uncertainty in this estimate follows
Power_law
Concept in mathematical physics
mechanics. Wu and Sprung also showed that the zeta-regularized functional determinant is the Riemann Xi-function ξ ( s ) ξ ( 0 ) = det ( H − s ( 1 − s ) +
Wu–Sprung_potential
British theoretical physicist and mathematician (1923–2020)
conjecture about the zeros of the zeta function. The primes 2, 3, 5, 7, 11, 13, 17, 19,... are described by the Riemann zeta function, and Dyson had previously
Freeman_Dyson
Integral transform generalizing both Laplace and Sumudu transforms
. {\displaystyle f(\zeta )\in A.} If the function f ( n ) ( t ) {\displaystyle f^{(n)}(t)} is the nth derivative of the function f ( t ) ∈ A {\displaystyle
Shehu_transform
{\displaystyle \operatorname {sgn}(x)} is the sign function. Other regulators, such as the zeta function regulator, may be used. The need for both a positive
Spectral_asymmetry
Method for numerical differential equations
β {\displaystyle \beta } and ζ {\displaystyle \zeta } are nondecreasing Lipschitz continuous functions: ∂ t β ( u ¯ ) − Δ ζ ( u ¯ ) = f {\displaystyle
Gradient discretisation method
Gradient_discretisation_method
Fifth letter of the Greek alphabet
extensions). in quantum field theory, it usually indicates the dimensional regularization parameter. In automata theory, it shows a transition that involves no
Epsilon
S2CID 119604277. Egger, Herbert; Engl, Heinz W. (2005). "Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis
PDE-constrained_optimization
Archaic letter of the Greek alphabet
of pair-wise exchange happened between Phoenician zayin and tsade: Greek zeta has the shape and position of zayin (𐤆) but the name and sound value of
San_(letter)
Mathematical function
geometric generating functions. Regularized theta lifts of harmonic Maass forms can be used to construct Arakelov Green functions for special divisors
Harmonic_Maass_form
Filling in missing entries of a matrix
completion problem is an application of matrix regularization which is a generalization of vector regularization. For example, in the low-rank matrix completion
Matrix_completion
Diacritic used in Latin alphabets
current name being zeta), itself derived from the Latin zeta, from Greek zêta, "the sixth letter of the Greek alphabet". Greek zêta is itself "borrowed
Cedilla
Theory for associative algebras over rings
the Hasse–Weil zeta function of a smooth proper variety over F p {\displaystyle \mathbb {F} _{p}} can be expressed using regularized determinants involving
Hochschild_homology
regression – redirects to Tikhonov regularization Ridit scoring Risk adjusted mortality rate Risk factor Risk function Risk perception Risk theory Risk–benefit
List_of_statistics_articles
Regression models accounting for possible errors in independent variables
0 ′ z t + σ 0 ζ t , {\displaystyle x_{t}^{*}=\pi _{0}'z_{t}+\sigma _{0}\zeta _{t},} where π0 and σ0 are (unknown) constant matrices, and ζt ⊥ zt. The
Errors-in-variables_model
Concept in statistics
model is the generalized Poisson distribution. Other possible models are the zeta distribution and the Zipf distribution. RR-VGLMs are VGLMs where a subset
Vector generalized linear model
Vector_generalized_linear_model
Romanian-American mathematician
commutative algebras (including differential graded commutative algebras); Zeta-regularized determinants of elliptic operators and implications to torsion invariants
Dan_Burghelea
Problem Margaret M. Robinson, American number theorist and expert on zeta functions Alvany Rocha, American specialist in Lie groups, computed characters
List_of_women_in_mathematics
Turkish-American theoretical physicist (1926–1994)
and Mass Splittings Within Su(3) and Su(2) Multiplets 1972 Value of the Zeta Parameter in Kl3 Decay from the Pion-Gauge Condition 1973 Kl3 and Pe3 Decays
Asım_Orhan_Barut
Constraint in loop quantum gravity
quantum operator in the loop representation. One introduces a lattice regularization procedure. we assume that space has been divided into tetrahedra Δ {\displaystyle
Hamiltonian_constraint_of_LQG
Method of hydrodynamics simulation
inter-particle averages amount to implicit dissipation, i.e. density regularization and numerical viscosity, respectively. Since the above discretization
Smoothed-particle hydrodynamics
Smoothed-particle_hydrodynamics
Technique in computational quantum field theory
strategy used in lattice calculations. In both cases a nonperturbative regularization and renormalization are used to try to construct effective theories
Light_front_quantization
Sounds and pronunciation of Ancient Greek
pronounced as a voiced [z] before voiced consonants. According to W.S. Allen, zeta ⟨ζ⟩ in Attic Greek likely represented the consonant cluster /sd/, phonetically
Ancient_Greek_phonology
Quantization procedure in quantum field theory
the interpretation of the wave functions, due to negative norm of the Pauli-Villars states introduced for regularization, becomes problematic. When the
Light-front quantization applications
Light-front_quantization_applications
Mathematical method in extremal graph theory
every μ > 0 {\displaystyle \mu >0} , there exists ζ > 0 {\displaystyle \zeta >0} and n 0 > 0 {\displaystyle n_{0}>0} so that the following holds. Suppose
Hypergraph_regularity_method
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ZETA FUNCTION-REGULARIZATION
ZETA FUNCTION-REGULARIZATION
ZETA FUNCTION-REGULARIZATION
ZETA FUNCTION-REGULARIZATION
ZETA FUNCTION-REGULARIZATION
ZETA FUNCTION-REGULARIZATION
ZETA FUNCTION-REGULARIZATION
ZETA FUNCTION-REGULARIZATION
ZETA FUNCTION-REGULARIZATION
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