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Difference between revisions of "Dirichlet formula"

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''for the number of divisors''
 
''for the number of divisors''
  
 
The asymptotic formula
 
The asymptotic formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032850/d0328501.png" /></td> </tr></table>
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$$\sum_{n\leq N}\tau(n)=N\ln N+(2C-1)N+O(\sqrt N),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032850/d0328502.png" /> is the number of divisors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032850/d0328503.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032850/d0328504.png" /> is the [[Euler constant|Euler constant]]. Obtained by P. Dirichlet in 1849; he noted that this sum is equal to the number of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032850/d0328505.png" /> with positive integer coordinates in the domain bounded by the hyperbola <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032850/d0328506.png" /> and the coordinate axes, i.e. equal to
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where $\tau(n)$ is the number of divisors of $n$ and $C$ is the [[Euler constant|Euler constant]]. 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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032850/d0328507.png" /></td> </tr></table>
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$$\left[\sqrt N\right]^2+2\sum_{x\leq\sqrt N}\left[\frac Nx\right]$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032850/d0328508.png" /> denotes the integer part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032850/d0328509.png" />.
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where $[\alpha]$ denotes the integer part of $\alpha$.
  
 
====References====
 
====References====

Revision as of 07:52, 9 August 2014

for the number of divisors

The asymptotic formula

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

where $\tau(n)$ is the number of divisors of $n$ and $C$ is the Euler constant. 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

See also Divisor problems.

How to Cite This Entry:
Dirichlet formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet_formula&oldid=32774
This article was adapted from an original article by A.F. Lavrik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article