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Dirichlet hyperbola method

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An example of the Dirichlet hyperbola method with and

In number theory, the Dirichlet hyperbola method is a technique to evaluate the sum

where is a multiplicative function. The first step is to find a pair of multiplicative functions and such that, using Dirichlet convolution, we have ; the sum then becomes

where the inner sum runs over all ordered pairs of positive integers such that . In the Cartesian plane, these pairs lie on a hyperbola, and when the double sum is fully expanded, there is a bijection between the terms of the sum and the lattice points in the first quadrant on the hyperbolas of the form , where runs over the integers : for each such point , the sum contains a term , and vice versa.

Let be a real number, not necessarily an integer, such that , and let . Then the lattice points can be split into three overlapping regions: one region is bounded by and , another region is bounded by and , and the third is bounded by and . In the diagram, the first region is the union of the blue and red regions, the second region is the union of the red and green, and the third region is the red. Note that this third region is the intersection of the first two regions. By the principle of inclusion and exclusion, the full sum is therefore the sum over the first region, plus the sum over the second region, minus the sum over the third region. This yields the formula



Let be the divisor-counting function, and let be its summatory function:

Computing naïvely requires factoring every integer in the interval ; an improvement can be made by using a modified Sieve of Eratosthenes, but this still requires time. Since admits the Dirichlet convolution , taking in (1) yields the formula

which simplifies to

which can be evaluated in operations.

The method also has theoretical applications: for example, Peter Gustav Lejeune Dirichlet introduced the technique in 1849 to obtain the estimate[1][2]

where is the Euler–Mascheroni constant.


  1. ^ Dirichlet, Peter Gustav Lejeune (1849). "Über die Bestimmung der mittleren Werthe in der Zahlentheorie". Abhandlungen der Königlich Preussischen Akademie der Wissenchaften (in German): 49–66 – via Gallica.
  2. ^ Tenenbaum, Gérald (2015-07-16). Introduction to Analytic and Probabilistic Number Theory. American Mathematical Soc. p. 44. ISBN 9780821898543.

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