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The distribution of values of \(a(n)\). (English) Zbl 0701.11039

Let \(a(n)\) be the number of non-isomorphic abelian groups of order \(n\). D. G. Kendall and R. A. Rankin [Q. J. Math. Oxf. 18, 197–208 (1947; Zbl 0031.15303)] were the first who investigated the number \(A_ k(x)\) of positive integers not exceeding \(x\) for which \(a(n)=k\) for fixed \(k\geq 0\). The author found asymptotic expansions for \(A_ k(x)\), where there is a clear distinction between even and odd values of \(k\). It seems that one can not improve the estimation for even \(k\). Assuming the Riemann hypothesis to be true, in this paper improvements for odd \(k\) are given. And now a clear distinction between the values \(k\equiv 3\pmod 6\) and \(k\equiv \pm 1\pmod 6\) is required.

MSC:

11N45 Asymptotic results on counting functions for algebraic and topological structures
20K01 Finite abelian groups

Citations:

Zbl 0031.15303
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References:

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