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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.

##### MSC:
 11N37 Asymptotic results on arithmetic functions 11N56 Rate of growth of arithmetic functions 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$
##### Keywords:
Riemann zeta-function; arithmetic sum; Laplace method