Number of divisors

2020 Mathematics Subject Classification: Primary: 11A25 [MSN][ZBL]

of a natural number $n$

The number of natural divisors of the number $n$. This arithmetic function is denoted by $\tau (n)$ or $d(n)$. The following formula holds: $$\tau (n)=(a_1+1)\cdots (a_k+1)$$ where $$n=p_1^{a_1}\cdots p_k^{a_k}$$ is the canonical expansion of $n$ into prime power factors. For prime numbers $p$, $\tau (p)=2$, but there exists an infinite sequence of $n$ for which $$\tau (n)\ge 2^{1-ϵ}\frac{\log n}{\log \log n},ϵ>0.$$

On the other hand, for all $ϵ>0$, $$\tau (n)=O(n^{ϵ}).$$

$\tau$ is a multiplicative arithmetic function and is equal to the number of points with natural coordinates on the hyperbola $xy=n$. The average value of $\tau (n)$ is given by Dirichlet's asymptotic formula (cf. Divisor problems).

The average value of the number of divisors was obtained by P. Dirichlet in 1849, in the form $$\sum _{n\le x}\tau (n)=x\log x+(2\gamma -1)x+O(\sqrt{x}).$$

The function ${\tau }_k(n)$, which is the number of solutions of the equation $n=x_1\cdots x_k$ in natural numbers $x_1,\dots ,x_k$, is a generalization of the function $\tau$.

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