Let x>e i6 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 «£JC 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.
ABSTRACT. This paper is concerned with estimating the number of positive integers up to some bound (which tends to infinity), such that they have a fixed number of prime divisors, and lie in a given arithmetic progression. We obtain estimates which are uniform in the number of prime divisors, and at the same time, in the modulus of the arithmetic progression. These estimates take the form of a fixed but arbitrary number of main terms, followed by an error term.
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