All Questions

For any positive integer $n$, let $\sigma(n)$ be the sum of all positive divisors of $n$. Clearly, $\sigma(n)\ge n\ge \varphi(n)$ for all $n\in\mathbb{Z}_{\geq 1}$, where $\varphi$ is Euler's totient ...

Let $f:\mathbb N\to \mathbb C$ be some arithmetical function. Define $\varphi_f(n)$ by the following formula: $$ \varphi_f(n)=\sum_{\substack{k\leq n \\ (k,n)=1}}f(k). $$ In other words, $\varphi_f(...

Let $\sigma_k(n)$ be the sum of the $k$-th powers of the positive divisors of $n$ and $\mu(n)$ be the Möbius function. As Arithmetic function - Wikipedia mentioned, there is an equation that $$\...

Suppose $f:\mathbb{N}\to [0,1]$ is a multiplicative function (i.e. $f(nm)=f(n)f(m)$ whenever $m$ and $n$ are coprime). Suppose $f$ has non-zero mean, which means $$ \lim_{N\to\infty}\frac{1}{N} \sum_{...

A function is called quasi-multiplicative by many authors if $f(m)f(n)=f(1)f(mn)$, a slight generalization of multiplicativity. (Basically a multiplicative function times a constant is a quasi-...

Carlitz showed necessary and sufficient conditions for an arithmetic function to be a linear combination of two multiplicative functions. He mentions the possibility of generalizing to $k$ ...