Divisor count summatory function

This article is about a function defined on positive reals (and in particular, natural numbers) obtained as the summatory function of an arithmetic function, namely divisor count function.
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Definition

Let $x$ be a positive real number. The divisor count summatory function of $x$ is defined as the summatory function for the divisor count function:

$x\mapsto \sum _{k\leq x}\sigma _{0}(k)$,

where the sum is over positive integers $k$, and $\sigma _{0}(k)$ is the number of positive integers dividing $k$.

This function can alternatively be defined as:

$\sum _{d\leq x}\left[{\frac {x}{d}}\right]$.

Here, the square braces represent the greatest integer function. In other words, while the earlier expression counts for each element less than or equal to $x$ the number of its divisors, the new expression counts for each possible divisor the number of multiples it has. Clearly, both expressions are equivalent.

Behavior

Growth

The divisor count function for a natural number $n$ grows as:

$n\log n+(2\gamma -1)n+O(n^{\theta })$,

where $\gamma$ is the Euler-Mascheroni constant, and $\theta$ is a (not completely determined) constant greater than $1/4$. The current best upper bound known for $\theta$ is $131/416$.