Asymptotics of an arithmetic sum. (Q1432466)

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.