Dirichlet formula

2020 Mathematics Subject Classification: Primary: 11N37 [MSN][ZBL]

for the number of divisors

The asymptotic formula

$$\sum_{n\leq N}\tau(n)=N\ln N+(2\gamma-1)N+O(\sqrt N),$$

where $\tau(n)$ is the number of divisors of $n$ and $\gamma$ is the Euler constant, $\gamma \approx 0.577$. Obtained by P. Dirichlet in 1849; he noted that this sum is equal to the number of points $(x,y)$ with positive integer coordinates in the domain bounded by the hyperbola $y=N/x$ and the coordinate axes, i.e. equal to

$$\left[\sqrt N\right]^2+2\sum_{x\leq\sqrt N}\left[\frac Nx\right]$$

where $[\alpha]$ denotes the integer part of $\alpha$.

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The formula implies that the average order of $\tau(n)$ is $\log n$.

See also Divisor problems.