Difference between revisions of "Dirichlet formula"
From Encyclopedia of Mathematics
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| + | {{TEX|done}}{{MSC|11N37}} | ||
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''for the number of divisors'' | ''for the number of divisors'' | ||
The asymptotic formula | The asymptotic formula | ||
| − | + | $$\sum_{n\leq N}\tau(n)=N\ln N+(2\gamma-1)N+O(\sqrt N),$$ | |
| − | where | + | 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 | + | where $[\alpha]$ denotes the integer part of $\alpha$. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> E.C. Titchmarsh, "The theory of the Riemann zeta-function" , Clarendon Press (1951)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> E.C. Titchmarsh, "The theory of the Riemann zeta-function" , Clarendon Press (1951)</TD></TR> | ||
| + | </table> | ||
====Comments==== | ====Comments==== | ||
| − | See also [[ | + | The formula implies that the [[Average order of an arithmetic function|average order]] of $\tau(n)$ is $\log n$. |
| + | |||
| + | See also [[Divisor problems]]. | ||
Latest revision as of 08:27, 30 December 2015
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 $\alpha$.
References
| [1] | E.C. Titchmarsh, "The theory of the Riemann zeta-function" , Clarendon Press (1951) |
Comments
The formula implies that the average order of $\tau(n)$ is $\log n$.
See also Divisor problems.
How to Cite This Entry:
Dirichlet formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet_formula&oldid=15641
Dirichlet formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet_formula&oldid=15641
This article was adapted from an original article by A.F. Lavrik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article