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Given an arithmetic function ''a''(''n''), let ''F''<sub>''a''</sub>(''s''), for complex ''s'', be the function defined by the corresponding [[Dirichlet series]] (where it [[Convergent series|converges]]):<ref>Hardy & Wright, § 17.6, show how the theory of generating functions can be constructed in a purely formal manner with no attention paid to convergence.</ref>
<math display="block"> F_a(s) := \sum_{n=1}^\infty \frac{a(n)}{n^s} .</math>
''F''<sub>''a''</sub>(''s'') is called a [[generating function]] of ''a''(''n''). The simplest such series, corresponding to the constant function ''a''(''n'') = 1 for all ''n'', is ''
The generating function of the Möbius function is the inverse of the zeta function:
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