Hardy Z-function

E239280

mathematical function object in analytic number theory special function

The Hardy Z-function is a real-valued function derived from the Riemann zeta function on the critical line, used extensively in the study of the distribution of its zeros and the Riemann Hypothesis.

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All labels observed (3)

Label	Occurrences
Hardy Z-function canonical	4
The Theory of Hardy’s Z-Function	1
ZFunction	1

How this entity was disambiguated

This entity first appeared as the object of triple T2171601 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Hardy Z-function
Context triple: [Riemann–Siegel formula, relatedTo, Hardy Z-function]

A. Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
B. Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
C. Jacobi elliptic functions
Jacobi elliptic functions are a family of doubly periodic complex functions that generalize trigonometric functions and play a central role in the theory of elliptic integrals and many areas of mathematical physics.
D. Weierstrass factorization theorem
The Weierstrass factorization theorem is a fundamental result in complex analysis that expresses any entire function as an infinite product determined by its zeros, generalizing the factorization of polynomials.
E. Hardy–Ramanujan asymptotic formula
The Hardy–Ramanujan asymptotic formula is a landmark result in number theory that gives an approximate expression for the partition function p(n), describing how the number of integer partitions of n grows rapidly with n.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Hardy Z-function
Target entity description: The Hardy Z-function is a real-valued function derived from the Riemann zeta function on the critical line, used extensively in the study of the distribution of its zeros and the Riemann Hypothesis.

A. Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
B. Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
C. Jacobi elliptic functions
Jacobi elliptic functions are a family of doubly periodic complex functions that generalize trigonometric functions and play a central role in the theory of elliptic integrals and many areas of mathematical physics.
D. Weierstrass factorization theorem
The Weierstrass factorization theorem is a fundamental result in complex analysis that expresses any entire function as an infinite product determined by its zeros, generalizing the factorization of polynomials.
E. Hardy–Ramanujan asymptotic formula
The Hardy–Ramanujan asymptotic formula is a landmark result in number theory that gives an approximate expression for the partition function p(n), describing how the number of integer partitions of n grows rapidly with n.
F. None of above. chosen

Statements (49)

Predicate	Object
instanceOf	mathematical function ⓘ object in analytic number theory ⓘ special function ⓘ
codomain	ℝ ⓘ
constructedFrom	Riemann zeta function ζ(s) ⓘ linked to: Riemann zeta function Riemann–Siegel theta function θ(t) ⓘ linked to: Riemann–Siegel theta function
definedOn	real line ⓘ
dependsOn	Riemann zeta function ⓘ
domain	t ∈ ℝ ⓘ
hasDefinition	Z(t) = e^{i\theta(t)} ζ(1/2 + it) ⓘ
hasProperty	Z(t) is real for real t ⓘ encodes values of the Riemann zeta function on the critical line ⓘ evenness or oddness depends on the chosen normalization of θ(t) ⓘ grows roughly like t^{1/4+o(1)} in magnitude under known bounds ⓘ inherits symmetries from the functional equation of ζ(s) ⓘ its large values are connected to extreme values of ζ(1/2 + it) ⓘ its real zeros correspond to zeros of the Riemann zeta function on the critical line ⓘ oscillatory behavior along the real axis ⓘ real-valued on the real line ⓘ sign changes of Z(t) indicate zeros of ζ(1/2 + it) ⓘ smooth except at points where ζ(1/2 + it) has poles or singularities (which do not occur on the critical line) ⓘ \|Z(t)\| = \|ζ(1/2 + it)\| for real t ⓘ
introducedBy	G. H. Hardy ⓘ
namedAfter	G. H. Hardy ⓘ
relatedTo	Gram points ⓘ Montgomery’s pair correlation conjecture (via zeros of ζ(1/2 + it)) ⓘ Riemann Hypothesis ⓘ linked to: Riemann hypothesis Riemann–Siegel theta function ⓘ critical line Re(s) = 1/2 of the Riemann zeta function ⓘ functional equation of the Riemann zeta function ⓘ moments of the Riemann zeta function on the critical line ⓘ zero counting function N(T) for the Riemann zeta function ⓘ
studiedIn	analytic number theory ⓘ computational number theory ⓘ
symbol	Z(t) ⓘ
usedFor	formulation of explicit formulae for zero counting ⓘ high-precision computation of ζ(1/2 + it) ⓘ locating zeros of ζ(s) on the critical line ⓘ numerical verification of the Riemann Hypothesis up to large heights ⓘ
usedIn	Gram’s law investigations ⓘ Riemann–Siegel formula ⓘ computational algorithms for ζ(1/2 + it) based on the Riemann–Siegel formula ⓘ computations of zeros of the Riemann zeta function on the critical line ⓘ investigation of the Riemann Hypothesis ⓘ statistical studies of zeros of the Riemann zeta function ⓘ study of zeros of the Riemann zeta function ⓘ verification of zero-free regions off the critical line by comparison ⓘ
variable	real variable t ⓘ
zeroCorrespondence	Z(t) = 0 if and only if ζ(1/2 + it) = 0 for real t ⓘ

How these facts were elicited

Referenced by (6)

Full triples — surface form annotated when it differs from this entity's canonical label.

Riemann–Siegel formula → relatedTo → Hardy Z-function ⓘ

WikiLambda → coreConcept → ZFunction ⓘ

linked to: Hardy Z-function

Riemann–Siegel theta function → argumentOf → Hardy Z-function ⓘ

Riemann–Siegel theta function → relatedTo → Hardy Z-function ⓘ

Gram points → relatedTo → Hardy Z-function ⓘ

Aleksandar Ivić → notableWork → The Theory of Hardy’s Z-Function ⓘ