Asymptotic expansion of the sum of the reciprocals of an arithmetic function
Let \[ S_r(x) := \sum_{1<n\leqslant x}{\left({1\over\pi(n)}\right)}^r, \] where \(r\geqslant 1\) is a fixed integer, and \(\pi(x)\) is the number of primes not exceeding \(x\). Asymptotic evaluation of \(S_1(x)\) was done by \textit{J.-M. De Koninck} and the reviewer [Topics in arithmetical functions. Amsterdam etc.: North-Holland (1980; Zbl 0442.10032)], \textit{L. Panaitopol} [Nieuw Arch. Wiskd. (5) 1, No. 1, 55--56 (2000; Zbl 0982.11003)], and the reviewer [Publ. Elektroteh. Fak., Univ. Beogr., Ser. Mat. 13, 85--88 (2002; Zbl 1053.11074)], who proved that \[ S_1(x) = {1\over2}\log^2x - \log x - \log\log x + C + \sum_{j=2}^m {k_j\over(j-1)\log^{j-1}x} + O\left({1\over\log^mx}\right),\leqno(1) \] where \(C\) is an absolute constant, \(m>1\) is fixed, and the \(k_j\)'s are defined recursively by \[ k_n + 1!k_{n-1} + \cdots + (n-1)!k_1 = n\cdot n!. \qquad(n \geqslant 1).\leqno(2) \] The authors have generalized (1)--(2) [Publ. Elektroteh. Fak., Univ. Beogr., Ser. Mat. 17, 45--51 (2006; Zbl 1188.11040)] by giving an analogous asymptotic formula for \(S_r(x)\). In the present paper a further generalization is obtained, when \(\pi(n)\) is replaced by an arithmetic function satisfying, for any fixed \(m\geqslant1\), \[ f(n) = {n^s\over \log^qn}\left(a_0 + {a_1\over\log n} + \cdots + {a_{m-1}\over\log^{m-1}m} + O\left({1\over\log^m n}\right)\right).\leqno(3) \] Clearly, by the prime number theorem, \(\pi(n)\) satisfies (3) with \(q=s=1\) and an arbitrary \(m\) (with (\(a_j = j!\))).
scientific article; zbMATH DE number 5554662
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