We introduce and analyze a general class of not necessarily bounded
multiplicative functions, examples of which include the function
, where
and where
counts the number of
distinct prime factors of
,
as well as the function
,
where
denotes the Fourier coefficients of a primitive holomorphic cusp form.
For this class of functions we show that after applying a
-trick,
their elements become orthogonal to polynomial nilsequences. The resulting functions
therefore have small uniformity norms of all orders by the Green–Tao–Ziegler inverse
theorem, a consequence that will be used in a separate paper in order to
asymptotically evaluate linear correlations of multiplicative functions from
our class. Our result generalizes work of Green and Tao on the Möbius
function.
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