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: Euler product expansions | Statement: [Multiplicative Number Theory, hasKeyTool, Euler product expansions]

Target entity: Euler product expansions
Context triple: [Multiplicative Number Theory, hasKeyTool, Euler product expansions]

• A. Euler product formula for the Riemann zeta function
  The Euler product formula for the Riemann zeta function is a fundamental identity in analytic number theory that expresses the zeta function as an infinite product over all prime numbers, revealing a deep connection between primes and the distribution of integers.
• B. Euler products for automorphic L-functions chosen
  Euler products for automorphic L-functions are infinite product expansions attached to automorphic representations that encode deep arithmetic information and generalize the classical Euler product of the Riemann zeta function to a broad class of L-functions in the Langlands program.
• C. Multiplicative Number Theory
  Multiplicative Number Theory is a branch of analytic number theory that studies arithmetic functions and prime number distributions through their multiplicative properties and associated Dirichlet series.
• D. Selberg–Delange method results
  Selberg–Delange method results are asymptotic formulas in analytic number theory that precisely describe the average order and distribution of multiplicative arithmetic functions using complex-analytic techniques.
• E. Dyson’s transform in number theory
  Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
• F. None of above.
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target predicate: hasKeyTool
Context triple: [Multiplicative Number Theory, hasKeyTool, Euler product expansions]

• A. hasKeyPass
  Indicates that an entity possesses or is granted a key-based pass that allows access or authorization to something.
• B. hasKeyInfrastructure
  Indicates that an entity possesses or is associated with essential infrastructure components critical for its operation or function.
• C. hasKeyMaterial
  Indicates that an entity possesses or contains the cryptographic key material required for encryption, decryption, or related security operations.
• D. hasKeyCapability chosen
  Indicates that an entity possesses an essential ability or function that is critical for performing a particular task or role.
• E. hasKeyUsage
  Indicates that an entity (such as a key or certificate) is associated with a specific intended purpose or allowed type of usage.
• F. None of above.