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Arithmetical function
the Dedekind psi function is the multiplicative function on the positive integers defined by ψ ( n ) = n ∏ p | n ( 1 + 1 p ) , {\displaystyle \psi (n)=n\prod
Dedekind_psi_function
Topics referred to by the same term
theory, Dedekind function can refer to any of three functions, all introduced by Richard Dedekind Dedekind eta function Dedekind psi function Dedekind zeta
Dedekind_function
Topics referred to by the same term
Psi function can refer, in mathematics, to the ordinal collapsing function ψ ( α ) {\displaystyle \psi (\alpha )} the Dedekind psi function ψ ( n ) {\displaystyle
Psi_function
German mathematician (1831–1916)
Richard Dedekind Dedekind cut Dedekind domain Dedekind eta function Dedekind-infinite set Dedekind number Dedekind psi function Dedekind sum Dedekind zeta
Richard_Dedekind
Topics referred to by the same term
AASHTO Psi Islands, in the Melchior Islands, Antarctica Chebyshev function Dedekind psi function Digamma function Polygamma functions Stream function, in
Psi
Number of integers coprime to and less than n
product of the first 120569 primes. Carmichael function (λ) Dedekind psi function (𝜓) Divisor function (σ) Duffin–Schaeffer conjecture Generalizations
Euler's_totient_function
Function whose domain is the positive integers
The Dedekind psi function, used in the theory of modular functions, is defined by the formula ψ ( n ) = n ∏ p | n ( 1 + 1 p ) . {\displaystyle \psi (n)=n\prod
Arithmetic_function
Arithmetical function
J_{k}(n)\sim {\frac {n^{k}}{\zeta (k+1)}}} . The Dedekind psi function is ψ ( n ) = J 2 ( n ) J 1 ( n ) {\displaystyle \psi (n)={\frac {J_{2}(n)}{J_{1}(n)}}} , and
Jordan's_totient_function
number Dedekind's problem Dedekind–Peano axioms Dedekind psi function Dedekind ring Dedekind sum Dedekind valuation Dedekind zeta function Dedekind–Hasse
List of things named after Richard Dedekind
List_of_things_named_after_Richard_Dedekind
Analytic function in mathematics
the Dirichlet L-functions and the Dedekind zeta function. For other related functions see the articles zeta function and L-function. The polylogarithm
Riemann_zeta_function
Mathematical concept
poles. More generally, the Riemann zeta function and the L-series can be replaced by the Dedekind zeta function of an algebraic number field or a Hecke
Explicit formulae for L-functions
Explicit_formulae_for_L-functions
Seventh letter in the Greek alphabet
lambda calculus. Mathematics, the Dirichlet eta function, Dedekind eta function, and Weierstrass eta function. In category theory, the unit of an adjunction
Eta
Mathematical function
Euler function, which is closely related to the Dedekind eta function. The Jacobi theta function may be written in terms of the Ramanujan theta function as:
Ramanujan_theta_function
Numbers obtained by adding the two previous ones
generating function of the Fibonacci numbers is given by the entire function F ( x ) = e φ x − e ψ x 5 {\displaystyle F(x)={\frac {e^{\varphi x}-e^{\psi x}}{\sqrt
Fibonacci_sequence
Plane algebraic curve
x with coefficients in Z[y], it has degree ψ(n), where ψ is the Dedekind psi function. Since Φn(x, y) = Φn(y, x), X0(n) is symmetrical around the line
Classical_modular_curve
Type of character in number theory
to construct a class of L-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which have
Hecke_character
Conjecture on zeros of the zeta function
extends the Riemann hypothesis to all Dedekind zeta functions of algebraic number fields. Since Dedekind zeta function for abelian extension of the rationals
Riemann_hypothesis
Number, approximately 1.46557
{\psi +1}{\psi ^{2}}}\\=\psi ^{2}-1&={\frac {\psi ^{7}+1}{\psi ^{7}-1}}\\\psi ^{4}-1&={\frac {\psi ^{2}+2}{\psi ^{2}-1}}\\\psi ^{6}-1&={\frac {\psi ^{2}+2}{\psi
Supergolden_ratio
Class of mathematical functions
parameterization using the sine function and its derivative the cosine function: ψ : R / 2 π Z → K , t ↦ ( sin t , cos t ) . {\displaystyle \psi :\mathbb {R} /2\pi
Weierstrass_elliptic_function
Standard system of axiomatic set theory
The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. However, the discovery of paradoxes in naive set theory,
Zermelo–Fraenkel_set_theory
Symbols for constants, special functions
size measure for analyses of variance the eta meson viscosity the Dedekind eta function energy conversion efficiency efficiency (physics) the pseudorapidity
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
Axiomatic set theories based on the principles of mathematical constructivism
Archimedean and Dedekind complete, if it exists at all, is in this way characterized uniquely, up to isomorphism. However, the existence of just function spaces
Constructive_set_theory
Pair of mathematical objects
x ) ∪ { 0 } . {\displaystyle \psi (x):=\sigma [x]\cup \{0\}=\varphi (x)\cup \{0\}.} By this, ψ ( x ) {\displaystyle \psi (x)} does always contain the number
Ordered_pair
System of mathematical set theory
{\text{Class}}(\psi ,\,n)\\\psi _{1}\land \psi _{2}:\;\;&\mathbf {return} \;\,{\text{Class}}(\psi _{1},\,n)\cap {\text{Class}}(\psi _{2},\,n);&&\\\;\;\;\;\
Von Neumann–Bernays–Gödel set theory
Von_Neumann–Bernays–Gödel_set_theory
Mathematical analysis
{T}}\vdash \phi \lor \psi } then also T ⊢ ϕ {\displaystyle {\mathsf {T}}\vdash \phi } or T ⊢ ψ {\displaystyle {\mathsf {T}}\vdash \psi } . Already in classical
Constructive_analysis
Statement that is taken to be true
numbers are uniquely picked out (up to isomorphism) by the properties of a Dedekind complete ordered field, meaning that any nonempty set of real numbers with
Axiom
Product of the first "n" prime numbers
Schoenfeld: Sharper bounds for the Chebyshev functions θ ( x ) {\displaystyle \theta (x)} and ψ ( x ) {\displaystyle \psi (x)} . II. Math. Comp. Vol. 34, No. 134
Primorial
Algorithm for computing greatest common divisors
extended by Richard Dedekind, who used Euclid's algorithm to study algebraic integers, a new general type of number. For example, Dedekind was the first to
Euclidean_algorithm
Integer having only small prime factors
( x , y ) {\displaystyle \Psi (x,y)} denote the number of y-smooth integers less than or equal to x (the de Bruijn function). If the smoothness bound
Smooth_number
Infinite integer series where the next number is the sum of the two preceding it
}(\phi ^{n}+\psi ^{n})x^{n}=\sum _{n=0}^{\infty }\phi ^{n}x^{n}+\sum _{n=0}^{\infty }\psi ^{n}x^{n}={\frac {1}{1-\phi x}}+{\frac {1}{1-\psi x}}=\Phi (x)}
Lucas_number
Particular class of sets which can be described entirely in terms of simpler sets
{\displaystyle \Psi } is the formula with the smallest Gödel number that can be used to define y {\displaystyle y} , and Ψ {\displaystyle \Psi } is different
Constructible_universe
System of mathematical set theory
{\displaystyle \forall a\in A\,\exists p\in P\,\psi (a,b,p)\,\land \,\forall p\in P\,\exists a\in A\,\psi (a,b,p)\,.} Given A {\displaystyle A} and collecting
Kripke–Platek_set_theory
Filtration of the Galois group of a local field extension
extension L of K. It is a generalization of the ramification theory of Dedekind domains. The structure of the set of extensions is known better when L/K
Ramification_group
Origin and evolution of the symbols used to write equations and formulas
_{k}p_{k}\,c\right)\psi (\mathbf {x} ,t)=i\hbar {\frac {\partial \psi (\mathbf {x} ,t)}{\partial t}}} where, ψ = ψ(x, t) is the wave function for the electron
History of mathematical notation
History_of_mathematical_notation
Form of mathematical proof
Augustus De Morgan, Charles Sanders Peirce, Giuseppe Peano, and Richard Dedekind. The simplest and most common form of mathematical induction infers that
Mathematical_induction
Count of permutations by cycles
( − 1 ) l ( 2 l + 1 ) ! ( 2 π z ) 2 l + 1 [ n 2 l + 1 ] {\displaystyle \Psi (z)=\ln z-{\frac {1}{2z}}-{\frac {1}{\pi z}}\sum _{n=1}^{\infty }{\frac {1}{n\cdot
Stirling numbers of the first kind
Stirling_numbers_of_the_first_kind
Number system extending the rational numbers
Napkin" (PDF). Retrieved 23 July 2025. Dedekind, Richard; Weber, Heinrich (2012), Theory of Algebraic Functions of One Variable, History of mathematics
P-adic_number
"The classical trilogarithm, algebraic K-theory of fields, and Dedekind zeta-functions" (PDF). Bull. AMS. pp. 155–162. Neumann, W.D. (2004). "Extended
Bloch_group
About simultaneous modular congruences
profinite integers, which is given as an inverse limit of all such maps. Dedekind's theorem on the linear independence of characters. Let M be a monoid and
Chinese_remainder_theorem
Increasing sequence of reduced fractions
..,N]=e^{\psi (N)}={\frac {1}{2}}\left(\prod _{r\in F_{N},0<r\leq 1/2}2\sin(\pi r)\right)^{2}} where ψ(N) is the second Chebyshev function. Since the
Farey_sequence
Concerned with the notion of stability in model theory
Alternatively, unrealized 1-types over a set A correspond to cuts (generalized Dedekind cuts, without the requirements that the two sets be non-empty and that
Stable_theory
Axiomatic set theory devised by W.V.O. Quine
f {\displaystyle {\mathsf {Inf}}} in this table only denotes "exist a Dedekind infinite set". All theories that do not have sufficient information to
New_Foundations
Mathematical system
2, together with an axiom schema of induction make up the usual Peano–Dedekind definition of N. Adding to these axioms any sort of axiom schema of induction
Second-order_arithmetic
Hungarian and American mathematician and physicist (1903–1957)
norm and is a vector ψ {\displaystyle \psi } which is such that V t ( ψ ) = ψ {\displaystyle V_{t}(\psi )=\psi } for all t {\displaystyle t} . This was
John_von_Neumann
Technique for proving independence results
function on I = [ 0 , 1 ] {\displaystyle I=[0,1]} . Real numbers in M [ G ] {\displaystyle M[G]} then correspond to Dedekind cuts of such functions,
Forcing_(mathematics)
Set of numbers used in the smoothsort algorithm
{\displaystyle L(n)=2{\frac {\varphi ^{n+1}-\psi ^{n+1}}{\varphi -\psi }}-1={\frac {2}{\sqrt {5}}}\left(\varphi ^{n+1}-\psi ^{n+1}\right)-1=2F(n+1)-1} where the
Leonardo_number
spacetime 2. Dedekind eta function, a weight 1/2 modular form 3. Eta meson, a neutral flavor meson with PC = –+ θ 1. Theta function 2. θc is the Cabbibo
Glossary_of_string_theory
Axiom set used in first-order logic
there exists a point b in r lying between X and Y. This is essentially the Dedekind cut construction, carried out in a way that avoids quantification over
Tarski's_axioms
Type of mathematical group
D_{F}} its discriminant, and ζ F {\displaystyle \zeta _{F}} its Dedekind zeta function. Let Γ O {\displaystyle \Gamma _{\mathcal {O}}} be the arithmetic
Arithmetic_Fuchsian_group
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DEDEKIND PSI-FUNCTION
DEDEKIND PSI-FUNCTION
DEDEKIND PSI-FUNCTION
DEDEKIND PSI-FUNCTION
DEDEKIND PSI-FUNCTION
DEDEKIND PSI-FUNCTION
DEDEKIND PSI-FUNCTION
DEDEKIND PSI-FUNCTION
DEDEKIND PSI-FUNCTION
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