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 $$\tau(n) \ge 2^{1-\epsilon} \frac{\log n}{\log\log n}\,,\ \ \epsilon > 0 \ .$$

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

$\tau$ 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 function $\tau_k(n)$, which is the number of solutions of the equation $n = x_1\cdots x_k$ in natural numbers $x_1,\ldots,x_k$, is a generalization of the function $\tau$.

References

[1]	I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)
[2]	K. Prachar, "Primzahlverteilung" , Springer (1957)

Comments

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

References

[a1]	G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. XVI