zbMATH — the first resource for mathematics

Asymptotics of an arithmetic sum. (English. Russian original) Zbl 1047.11092
Dokl. Math. 63, No. 1, 48-51 (2001); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 376, No. 3, 307-310 (2001).
Let \(d_k(n)\) denote the number of ways \(n\) can be written as a product of \(k\) factors, so that \(d_k(n)\) is generated by \(\zeta^k(s)\), the \(k\)th power of the Riemann zeta-function. The author is interested in the summatory function of \(d_k(n)\) when \(k\) is not fixed. Using Perron’s formula he proves the following theorem: Let \(C_1(\log\log x)^\beta < k < C_2\log^\alpha x\), where \(C_1 > 0, C_2 >0, 0 < \alpha < 2/3, \beta > 6\) are constants. Then \[ \sum_{n\leq x}d_k(n) = x{(\log x)^{k-1}\over(k-1)!} \text{ e}^{\gamma {k^2\over\log x}}\left(1 + O(k^{-\rho_0})\right), \] where \(\gamma = -\Gamma'(1)\) is Euler’s constant, and \(\rho_0 > 0\) is a constant.

11N37 Asymptotic results on arithmetic functions
11N56 Rate of growth of arithmetic functions
11M06 \(\zeta (s)\) and \(L(s, \chi)\)