The Probability that the Number of Groups of Squarefree Order is Two More than a Fixed Prime | Zendy

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The Probability that the Number of Groups of Squarefree Order is Two More than a Fixed Prime

Author(s) -

Spiro Claudia A.

Publication year - 1990

Publication title -

proceedings of the london mathematical society

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.899

H-Index - 65

eISSN - 1460-244X

pISSN - 0024-6115

DOI - 10.1112/plms/s3-60.3.444

Subject(s) - mathematics , square free integer , combinatorics , isomorphism (crystallography) , order (exchange) , prime (order theory) , asymptotic formula , prime number , prime factor , discrete mathematics , crystallography , chemistry , finance , crystal structure , economics

Let x > e 16 be real, and let k and n denote positive integers. Signify by g ( n ) the number of (isomorphism classes of) groups of order n , and write Q k ( x ) for the number of squarefree positive integers n ⩽ x with g ( n ) = k . We prove an asymptotic formula for Q k ( x ) when k − 2 is prime and also satisfies a side condition. The first seven values of k such that our theorem gives an asymptotic formula for Q k ( x ) are 7, 19, 31, 49, 73, 91, and 103.

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