Questions tagged [multiplicative-function]

In number theory, a multiplicative function is a function defined on positive integers such that f(ab)=f(a)f(b) for a,b coprime. E.g. Euler's totient function, sum of divisors and number of divisors are multiplicative functions.

188 questions
50 views

A strange multiplicative function

I have a function that looks like $$f_n(i) = \max\left(\left\lfloor \frac{i-n}{2} \right \rfloor + 1, 0\right)$$ and I would like to write it as a nice arithmetic function. To give an idea, another ...
88 views

Sum of divisors, a congruence

Let $\sigma_r(n)=\sum_{d|n}d^r$ where the sum is over all the integers $d=1,\dots,n$ which divide $n$. I am conjecturing $$\sum_{m=1}^{p-1}m^2 \sigma_3(m)\sigma_3(p-m)\not\equiv 0\pmod{p^2}$$ for ...
107 views

50 views

Cardinality of the set of either completely multiplicative or multiplicative functions defined on $\mathbb N$ and which take values in $\mathbb N$

Suppose that $S$ is a set of functions such that $f \in S$ if and only if $f$ is either multiplicative or completely multiplicative and every $f \in S$ is defined on the set of natural numbers and ...