On the missing \(\log\) factor
This is a survey paper, which is concerned with the problem of establishing upper bounds for sums \[ \sum_{n \leq x} \mu(n) g(x,n). \] Here $\mu$ denotes the Möbius function, and $g$ is a regular function. The author covers the history of the problem and many classical and recent results, with an emphasis on the connection to functional analysis which comes from the fact that the Möbius function can be defined as the convolution inverse of the function $\mathbf{1}$. The writing style is pleasant and leisurely, providing easy access to the topic. The ``missing log factor'' in the title refers to Meissel's equations \[ \sum_{n \leq x} \mu(x) [x/n] = 1 \] and \[ \sum_{n \leq x} \mu(n) \{x/n\} = - 1 + x \sum_{n \leq x} \frac{\mu(n)}{n}, \] where $[\cdot]$ and $\{ \cdot\}$ denote the integer resp. fractional part. In the second equation the left-hand side can be trivially estimated by $n$, and the right-hand side by $n \log n$ -- a missing log factor on the left! For the entire collection see [Zbl 1397.37007].
scientific article; zbMATH DE number 7005157