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Dirichlet formula

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for the number of divisors

The asymptotic formula

where is the number of divisors of and is the Euler constant. Obtained by P. Dirichlet in 1849; he noted that this sum is equal to the number of points with positive integer coordinates in the domain bounded by the hyperbola and the coordinate axes, i.e. equal to

where denotes the integer part of .

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