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Let , , and be multiplicative functions such that , where denotes the convolution of and . The Dirichlet hyperbola method (typically shortened to hyperbola method) is a way to calculate
by using the fact that :
Note that, since
, not both of and can be larger than . The Dirichlet hyperbola method follows from this fact as well as the inclusion-exclusion principle.
This method for calculating
is advantageous when the sums in terms of and are easier to handle and when
is relatively small for most
.
As an example, the sum
will be calculated using the Dirichlet hyperbola method.
Note that . Thus:
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