Generalised sum of divisors.

Question

Let $$F(x,k)=\sum_{a_{1}a_{2}...a_{k}\le x}1$$

Find asymptotic formula for $F(x,k)$

For example it is known. $$F(x,2)=\sum_{n\le x}\tau(x)=x\log x+(2\gamma-1)x+O(x^{\frac{1}{2}})$$ The proof of this follows from Dirichlet hyperbola method which uses inclusion-exclusion principle and the fact that if product of two numbers is no grater than $x$, then at least one of them is no grater than $x^{\frac{1}{2}}$.

Here is my question: Is it possible to generalise this method. There should be also used a inclusion-exclusion principle and a new fact that if product of $k$ numbers doesn't exceed $x$ then at least one of them is no grater than $x^{\frac{1}{k}}$

Answer 1

You can just note that $$\sum_{a_1\dots a_k\leq x}1 =\sum_{a_k\leq x}\sum_{a_1\dots a_{k-1}\leq x/a_k}1$$ which implies the recurrence formula $$F(x,k)=\sum_{a\leq x}F(x/a,k-1).$$ Thus you can derive the asymptotic behaviour of $F(x,k)$ from that of $F(x,k-1)$. For example, considering only the dominating term in the asymptotic formula for $F(x,k)$, we have $$F(x,3)=\sum_{a\leq x}\frac xa\log(x/a)+O(x\log x) =\frac12x\log^2x+O(x\log x).$$ Similarly $$F(x,4)\sim\frac16x\log^3x$$ and so on.

Answer 2

This is a classical topic and is treated in the standard references like Titchmarsh (theory of Riemann Zeta, chapter 12) etc; there is always an asymptotic formula of the form $xP_n(\log x) + R(x), \deg P_n = n-1, R(x)=O(x^{\alpha_n+\epsilon}), \frac{n-1}{2n} \le \alpha_n \le \frac{n-1}{n+1}$.

$P_n(x)$ is the residue at $1$ of $F(w)=\zeta(w)^n\frac{x^w}{w}$ which has a pole of order $n$ there.

The values of $\alpha_n$ (conjectured to be at their lower end, namely $\frac{n-1}{2n}$) are part of the circle of ideas around the Lindelof Hypothesis and Exponent Pairs and they are considered among the hardest problems in Analytic number theory (for example LH is equivalent to all $\alpha_n \le \frac{1}{2}$) while even the simplest case $\alpha_2 =\frac{1}{4}$is not known,the best estimate today still being above $.3$ after 100+ years of intensive research and tons of papers improving it little by little (Voronoi proved the now classical $\alpha_2 \le \frac{1}{3}$ in 1902, while Hardy proved the Omega results in 1914-5