Numbers which are only orders of abelian or nilpotent groups (Q6072955)

A number \(n\) is called \textit{cyclic} if every group of order \(n\) is cyclic, \textit{abelian} if every group of order \(n\) is abelian, and \textit{nilpotent} if every group of order \(n\) is nilpotent. A number \(n\) is \textit{strictly abelian} if every group of order \(n\) is abelian and there is at least a group of order \(n\) that is not cyclic. Similarly, \(n\) is \textit{strictly nilpotent} if any group of order \(n\) is nilpotent and there is at least one group of order \(n\) which is not abelian. Let \(C(x), A(x)\), and \(N(x)\) be the counting functions for cyclic, abelian, and nilpotent numbers, respectively. Then \(A(x) - C(x)\) and \(N(x) - A(x)\) are the counting functions for strictly abelian and strictly nilpotent numbers, respectively. Work of Erdős, Mays, Scourfield, Narlikar and Srinivasan have found asymptotic formulas for \(C(x), A(x), N(x), A(x)-C(x),\) and \(N(x) - A(x)\). \textit{P. Pollack} [Proc. Am. Math. Soc. 150, No. 2, 515--524 (2022; Zbl 1503.11130)] (improving on Begunts) found an asymptotic series expansion for \(C(x)\). In this paper, the author extends on Pollack's work to get asymptotic series expansions for \(A(x) - C(x)\) and \(N(x) - A(x)\). In particular they prove two main theorems: Theorem 1. For any positive integer \(N\), there is a sequence of real numbers \(b_0 = 1, b_1, b_2, \ldots, b_N\) such that \[A(x) - C(x) = \frac{x e^{-\gamma}}{\log_2{x}(\log_3{x})^2}\left(\sum_{k=0}^N \frac{b_k}{(\log_3{x})^k}\right)+O_N\left(\frac{x}{\log_2{x}(\log_3{x})^{N+3}}\right),\] where \(\log_2{x} = \log\log{x}, \log_3{x} = \log\log\log{x}\), and the constants \(b_k\) are determined as follows. Let \(C_k\) be the coefficients of the series expansion of \(\Gamma(1+w)\) about \(w = 0\). Let \(c_0 = 1, c_1 = -\gamma\), \(c_2 = \gamma^2 + \frac{1}{12}\pi^2\), and in general \(c_k\) be determined by the formal relation \[c_0 + c_1z + c_2z^2 + \cdots = \exp \left(0!C_1z + 1!C_2z^2 + 2!C_3z^3+\cdots\right),\] Then \[b_k = \sum_{\substack{i+k=k\\ i,j\ge0}} j!c_iC_j.\] Theorem 2. For any positive integer \(N\), there is a sequence of real numbers \(d_0 = 1, d_1, d_2, \ldots, d_N\) such that \[N(x) - A(x) = \frac{x e^{-\gamma}}{(\log_2{x})^2(\log_3{x})^2}\left(\sum_{k=0}^N \frac{d_k}{(\log_3{x})^k}\right)+O_N\left(\frac{x}{\log_2{x}(\log_3{x})^{N+3}}\right),\] where \(\log_2{x} = \log\log{x}, \log_3{x} = \log\log\log{x}\), and the constants \(d_k\) are determined as follows. Let \(c_k\) be defined as above, and \(D_k\) be the coefficients of the series expansion of \(\Gamma(2+w)\) about \(w = 0\). Then \[d_k = \sum_{\substack{i+k=k\\ i,j\ge0}} j!c_iD_j.\] The proofs involve intricate work with the Selberg sieve, the prime number theorem (with a good error term), and precise estimates on sums of reciprocals of primes in arithmetic progressions. Furthermore, the author had to be careful to account for numbers that are not squarefree.