Gauss sum
E662760
A Gauss sum is a finite exponential sum, often involving characters such as the Legendre symbol, that plays a central role in number theory and the study of quadratic and higher-order residues.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Gauss sums | 5 |
| Gauss sum canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7420179 — 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: Gauss sum Context triple: [Legendre symbol, relatedConcept, Gauss sum]
-
A.
Ramanujan’s sum
Ramanujan’s sum is a number-theoretic function introduced by Srinivasa Ramanujan, expressing certain periodic arithmetic functions as finite trigonometric sums over primitive roots of unity.
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B.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
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C.
Gauss multiplication formula
The Gauss multiplication formula is a classical identity in complex analysis that expresses the gamma function of a multiple of a variable as a product of gamma functions evaluated at shifted fractions of that variable.
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D.
Poisson summation formula
The Poisson summation formula is a fundamental result in harmonic analysis that links sums of a function over the integers to sums of its Fourier transform, with deep applications in number theory, signal processing, and physics.
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E.
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.
- 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: Gauss sum Target entity description: A Gauss sum is a finite exponential sum, often involving characters such as the Legendre symbol, that plays a central role in number theory and the study of quadratic and higher-order residues.
-
A.
Ramanujan’s sum
Ramanujan’s sum is a number-theoretic function introduced by Srinivasa Ramanujan, expressing certain periodic arithmetic functions as finite trigonometric sums over primitive roots of unity.
-
B.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
C.
Gauss multiplication formula
The Gauss multiplication formula is a classical identity in complex analysis that expresses the gamma function of a multiple of a variable as a product of gamma functions evaluated at shifted fractions of that variable.
-
D.
Poisson summation formula
The Poisson summation formula is a fundamental result in harmonic analysis that links sums of a function over the integers to sums of its Fourier transform, with deep applications in number theory, signal processing, and physics.
-
E.
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.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
exponential sum
ⓘ
mathematical concept ⓘ |
| appearsIn |
Gauss’s Disquisitiones Arithmeticae
ⓘ
linked to:
Disquisitiones Arithmeticae
|
| context |
finite fields F_q
ⓘ
integers modulo n ⓘ |
| definitionInvolves |
Dirichlet character
ⓘ
linked to:
Dirichlet characters
Legendre symbol ⓘ additive character ⓘ complex exponential ⓘ finite sum ⓘ multiplicative character ⓘ roots of unity ⓘ |
| field | number theory ⓘ |
| hasMagnitudeProperty | absolute value often equals square root of modulus ⓘ |
| hasSpecialCase |
Jacobi sum
ⓘ
linked to:
Jacobi sums
Kloosterman sum ⓘ cubic Gauss sum ⓘ quadratic Gauss sum ⓘ |
| namedAfter | Carl Friedrich Gauss ⓘ |
| property |
complex-valued
ⓘ
depends on character ⓘ depends on modulus ⓘ finite length ⓘ |
| relatedTo |
Gauss reciprocity law
ⓘ
linked to:
quadratic reciprocity law
Gauss sums over finite fields ⓘ Gaussian periods ⓘ Ramanujan sum ⓘ
linked to:
Ramanujan’s sum
Weil bound ⓘ
linked to:
Weil conjectures
quadratic reciprocity ⓘ |
| specialCaseOf |
character sum
ⓘ
exponential character sum ⓘ |
| usedIn |
L-function theory
ⓘ
Weil conjectures ⓘ algebraic number theory ⓘ analytic number theory ⓘ class field theory ⓘ coding theory ⓘ cryptography ⓘ finite field theory ⓘ higher-order residue theory ⓘ local class field theory ⓘ quadratic residue theory ⓘ representation theory of finite groups ⓘ |
| usedToCompute |
Fourier transform of characters on finite abelian groups
ⓘ
exponential sums over finite fields ⓘ number of solutions of polynomial congruences ⓘ |
| usedToProve |
non-vanishing of L-functions at certain points
ⓘ
properties of Dirichlet L-functions ⓘ quadratic reciprocity law ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Gauss sum Description of subject: A Gauss sum is a finite exponential sum, often involving characters such as the Legendre symbol, that plays a central role in number theory and the study of quadratic and higher-order residues.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Gauss sum
linked to: Gauss sum
linked to: Gauss sum
linked to: Gauss sum
linked to: Gauss sum