Triple

Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).

Phrase: Dirichlet inverse | Statement: [Dirichlet convolution, inverseOperationName, Dirichlet inverse]

Target entity: Dirichlet inverse
Context triple: [Dirichlet convolution, inverseOperationName, Dirichlet inverse]

• A. Dirichlet convolution
  Dirichlet convolution is a binary operation on arithmetic functions that combines them via summation over divisors and plays a central role in multiplicative number theory and Dirichlet series.
• B. Möbius function
  The Möbius function is a multiplicative arithmetic function in number theory that assigns values based on the prime factorization of integers and plays a central role in inversion formulas and the study of prime distribution.
• C. Dirichlet hyperbola method
  The Dirichlet hyperbola method is a technique in analytic number theory used to estimate sums of arithmetic functions by splitting double sums along a hyperbola to obtain asymptotic formulas.
• D. Liouville function
  The Liouville function is a completely multiplicative arithmetic function that assigns values based on the parity of the total number of prime factors of an integer, playing a key role in analytic number theory and the study of prime distribution.
• E. Dirichlet series
  A Dirichlet series is an infinite series of the form ∑ aₙ/nˢ, fundamental in analytic number theory for studying arithmetic functions and L-functions.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Dirichlet inverse
Target entity description: The Dirichlet inverse of an arithmetic function is another arithmetic function that, when combined with the original via Dirichlet convolution, yields the identity function concentrated at 1.

• A. Dirichlet convolution
  Dirichlet convolution is a binary operation on arithmetic functions that combines them via summation over divisors and plays a central role in multiplicative number theory and Dirichlet series.
• B. Möbius function
  The Möbius function is a multiplicative arithmetic function in number theory that assigns values based on the prime factorization of integers and plays a central role in inversion formulas and the study of prime distribution.
• C. Dirichlet hyperbola method
  The Dirichlet hyperbola method is a technique in analytic number theory used to estimate sums of arithmetic functions by splitting double sums along a hyperbola to obtain asymptotic formulas.
• D. Liouville function
  The Liouville function is a completely multiplicative arithmetic function that assigns values based on the parity of the total number of prime factors of an integer, playing a key role in analytic number theory and the study of prime distribution.
• E. Dirichlet series
  A Dirichlet series is an infinite series of the form ∑ aₙ/nˢ, fundamental in analytic number theory for studying arithmetic functions and L-functions.
• F. None of above. chosen

Target predicate: inverseOperationName
Context triple: [Dirichlet convolution, inverseOperationName, Dirichlet inverse]

• A. opposedOperation
  Indicates that one operation is in conflict with, counters, or works against another operation.
• B. inverseConvolution chosen
  Indicates the operation that reverses or undoes a prior convolution, recovering an original signal or function from its convolved form.
• C. isInverseOf
  Indicates that one relation reverses the direction of another, so that if the original relates A to B, its inverse relates B to A.
• D. inverseImage
  Indicates the mapping from a set of outputs back to all inputs that are related to those outputs under a given function or relation.
• E. inOperation
  Indicates that an entity (such as a system, device, or facility) is currently active, functioning, and performing its intended operation.
• F. None of above.